hcju/proofwikips huggingface dataset:
Heine-Borel Theorem/Necessary Condition
Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subseteq X$ be closed and bounded . Then $K$ is a compact subset.
Heine-Borel Theorem/Sufficient Condition
Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subset X$ be a compact subset. Then $K$ is closed and bounded.
Maximum Number of Arcs in Digraph
Let $D_n$ be a digraph of order $n$ such that $n \ge 1$. Let $D_n$ have the greatest number of arcs of all digraphs of order $n$. The number of arcs in $D$ is given by: :$\size {D_n} = n \paren {n - 1}$
Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p: x \ne 0$. Then: :$\exists v \in \Z: \norm x_p = p^{-v}$ === Lemma === {{:Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma}}
Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Let $\sequence {x_n}$ be a Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$ such that $\sequence {x_n}$ does not converge to $0$. Then: :$\exists v \in \Z: \displaystyle \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
Cycle Graph is Hamiltonian
Let $n \in \Z$ be an integer such that $n \ge 3$. Let $C_n$ denote the cycle graph of order $n$. Then $C_n$ is Hamiltonian.
Maximum Degree of Vertex in Simple Graph
Let $G = \struct {V, E}$ be a simple graph. Let $\card V$ denote the order of $G$. Then no vertex of $G$ has a degree higher than $\card V - 1$.
Degrees of Vertices determine Order and Size of Graph
Let $G = \struct {V, E}$ be a simple graph. Let the degrees of each of the vertices of $G$ be given. Then it is possible to determine both the order $\card V$ and size $\card E$ of $G$.
Order and Size of Graph do not determine Degrees of Vertices
Let $G = \struct {V, E}$ be a simple graph. Let both the order $\card V$ and size $\card E$ of $G$ be given. Then it is not always possible to determine the degrees of each of the vertices of $G$.
Size of Regular Graph in terms of Degree and Order
Let $G = \struct {V, E}$ be an $r$-regular graph of order $p$. Let $q$ denote the size of $G$. Then: :$q = \dfrac {p r} 2$ when such an $r$-regular graph exists. If an $r$-regular graph of order $p$ does exist, then $p r$ is an even integer.
Same Degrees of Vertices does not imply Graph Isomorphism
Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs such that: :$\card {\map V G} = \card {\map V H}$ where $\card {\map V G}$ denotes the order of $G$. Let $\phi: G \to H$ be a mapping which preserves the degrees of the vertices: :$\forall v \in \map V G: \map {\deg_H} {\map \phi v} = \map {\deg_G} v$ Then it is not necessarily the case that $\phi$ is an isomorphism.
Isomorphism Classes for Order 2 Simple Graphs
There are $2$ equivalence classes for simple graphs of order $2$ under graph isomorphism: :the edgeless graph of order $2$ and :the complete graph of order $2$.
Isomorphism Classes for Order 4 Size 3 Simple Graphs
There are $3$ equivalence classes for simple graphs of order $4$ and size $3$ under isomorphism: :400px
Differential Equation governing First-Order Reaction
Let a substance decompose spontaneously in a '''first-order reaction'''. The differential equation which governs this reaction is given by: :$-\dfrac {\d x} {\d t} = k x$ where: :$x$ determines the quantity of substance at time $t$. :$k \in \R_{>0}$.
Formula for Radiocarbon Dating
Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined. Let $t$ years be the age of $Q$ which is to be determined. Let $r$ denote the ratio of the quantity of carbon-14 remaining in $Q$ after time $t$ to the quantity of carbon-14 in $Q$ at the time of its death. Then the number of years that have elapsed since the death of $Q$ is given by: :$t = -8060 \ln r$
Leigh.Samphier/Sandbox/Representative of P-adic Sum
Let $p$ be any prime number. Let $\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. and $\norm {\,\cdot\,}_p$ denotes the p-adic norm on the rationals $\Q$. Let $x, y$ be any left cosets of $\Q_p$. Let $\sequence{x_n}$ and $\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\sequence{x_n + y_n}$ is a repesentative of $x + y$
Leigh.Samphier/Sandbox/Representative of P-adic Product
Let $p$ be any prime number. Let $\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. and $\norm {\,\cdot\,}_p$ denotes the p-adic norm on the rationals $\Q$. Let $x, y$ be any left cosets of $\Q_p$. Let $\sequence{x_n}$ and $\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\sequence{x_n y_n}$ is a repesentative of $x y$
Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring
Let $\struct {R, \norm {\, \cdot \,} }$ be a complete normed division ring. Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then: :$\struct {R, \norm {\, \cdot \,} }$ is a completion of $\struct {S, \norm {\, \cdot \,}}$ where the inclusion mapping $i : S \to R$ is the required distance-preserving ring monomorphism.
Leigh.Samphier/Sandbox/Distance-Preserving Homomorphism Preserves Norm
Let $\struct {R_1, \norm {\, \cdot \,}_1 }, \struct {R_2, \norm {\, \cdot \,}_2 }$ be normed division rings. Let $\phi: R_1 \to R_2$ be a distance-preserving ring homomorphism. Then: :$\forall x \in R_1 : \norm{\map \phi x}_2 = \norm x_1$
Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring
Let $\struct {R_1, \norm {\, \cdot \,}_1 }, \struct {R_2, \norm {\, \cdot \,}_2 }$ be normed division rings. Let $\struct {R_2, \norm {\, \cdot \,}_2 }$ be a completion of $\struct {R_1, \norm {\, \cdot \,}_1 }$ with distance-preserving ring monomorphism $\phi: R_1 \to R_2$. Then for all $x \in R_2$, there exists a sequence $\sequence{x_n}$ in $R_1$: :$x = \displaystyle \lim_{n \mathop \to \infty} \map \phi {x_n}$
Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion
Let $\struct {R_1, \norm {\, \cdot \,}_1 }$ be a normed division ring. Let $\struct {R_2, \norm {\, \cdot \,}_2 }$ be a normed division ring completion of $\struct {R_1, \norm {\, \cdot \,}_1 }$ with distance-preserving ring monomorphism $\phi: R_1 \to R_2$. Then for all $x \in R_2$, there exists a sequence $\sequence{x_n}$ in $R_1$: :$x = \displaystyle \lim_{n \mathop \to \infty} \map \phi {x_n}$ and :$\norm x_2 = \displaystyle \lim_{n \mathop \to \infty} \norm {x_n}_1$
Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary
Let $\struct {R, \norm {\, \cdot \,} }$ be a complete normed division ring. Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$: :$x = \displaystyle \lim_{n \mathop \to \infty} x_n$ and :$\norm x = \displaystyle \lim_{n \mathop \to \infty} \norm {x_n}$
Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {S, \norm {\, \cdot \,}_S}$ be a normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then the inclusion mapping $i : S \to R$ is a distance-preserving ring monomorphism.
Period of Reciprocal of Prime
Consider the decimal expansion of the reciprocal $\dfrac 1 p$ of a prime $p$. If $p \nmid a$, the decimal expansion of $\dfrac 1 p$ is periodic in base $a$ and its period of recurrence is the order of $a$ modulo $p$. If $p \divides a$, the decimal expansion of $\dfrac 1 p$ in base $a$ terminates.
Maximum Period of Reciprocal of Prime
Let $p$ be a prime number such that $p$ is not a divisor of $10$. The period of recurrence of the reciprocal of $p$ when expressed in decimal notation is less than or equal to $p - 1$.
Maximum Abscissa for Loop of Folium of Descartes
Consider the folium of Descartes defined in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$ :500px The point on the loop at which the $x$ value is at a maximum occurs when $t = \sqrt [3] {\dfrac 1 2}$, corresponding to the point $P$ defined as: :$P = \tuple {2^{2/3} a, 2^{1/3} a}$
Condition for Denesting of Square Root/Lemma
Let $a, b, c, d \in \Q_{\ge 0}$. Suppose $\sqrt b \notin \Q$. Then: :$\sqrt {a + \sqrt b} = \sqrt {c + \sqrt d} \implies a = c, b = d$
First Order ODE/y' + y = 0
The first order ODE: :$\dfrac {\d y} {\d x} + y = 0$ has the general solution: :$y = C e^{-x}$ where $C$ is an arbitrary constant.
Order-Preserving Mapping Not Always Semilattice Homomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices. Let $\preceq_1$ be the ordering on $S$ defined by: :$a \preceq_1 b \iff \paren {a \circ b} = b$ Let $\preceq_2$ be the ordering on $T$ defined by: :$x \preceq_2 y \iff \paren {x * y} = y$ Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be an order-preserving mapping. Then: : $\phi: \struct {S, \circ} \to \struct {T, *}$ may not be a semilattice homomorphism
Factorial as Product of Consecutive Factorials/Lemma 2
Let $n \in \N$. Then $\paren {2 n - 2}! \, \paren {2 n - 1}! > \paren {3 n - 1}!$ for all $n \ge 7$.
Factorial as Product of Consecutive Factorials/Lemma 1
Let $n \in \N$. Then $\paren {2 n - 1}! \, \paren {2 n}! > \paren {3 n - 1}!$ for all $n > 1$.
Entire Function with Bounded Real Part is Constant
Let $f : \C \to \C$ be an entire function.  Let the real part of $f$ be bounded. That is, there exists a positive real number $M$ such that:  :$\cmod {\map \Re {\map f z} } < M$ for all $z \in \C$, where $\map \Re {\map f z}$ denotes the real part of $\map f z$. Then $f$ is constant.
Mass of Mole of Isotope of Element
Let $S$ be a substance made up entirely of a particular isotope $Q$ of a particular element. Let one atom of $Q$ contain $n$ neutrons and $p$ protons. Then one mole of $S$ has a mass of approximately $n + p$ grams.
Mass of Mole of Substance
Let $S$ be a substance with molecular weight $W_S$. Then one mole of $S$ has a mass of $W_S$ grams.
Linear Second Order ODE/y'' = y'
The second order ODE: :$(1): \quad y'' = y'$ has the general solution: :$y = A_1 e^x + A_2$
First Order ODE/x y' = 2 y
The first order ODE: :$x y' = 2 y$ has the general solution: :$y = C x^2$ where $C$ is an arbitrary constant.
Singleton in Normed Vector Space is Closed
Let $X$ be a vector space over $\R$ or $\C$. Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$ be a singleton. Then $x$ is closed.
Singleton Set is Nowhere Dense in Rational Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then every singleton subset of $\Q$ is nowhere dense in $\struct {\Q, \tau_d}$.
Real Function of Two Variables represents Surface in Cartesian 3-Space
Let $S$ and $T$ be subsets of the set of real numbers $\R$. Let $f: S \times T \to \R$ be a real function of two variables. Then the locus of $f$ describes a surface embedded in the Cartesian space $\R^3$.
Cauchy's Lemma (Number Theory)
Let $a$ and $b$ be odd positive integers.  Suppose $a$ and $b$ satisfy:  :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$ Then there exist non-negative integers $s, t, u, v$ such that: {{begin-eqn}} {{eqn | l = a       | r = s^2 + t^2 + u^2 + v^2 }} {{eqn | l = b       | r = s + t + u + v }} {{end-eqn}}
Finite Subset of Normed Vector Space is Closed
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $F \subseteq X$ be finite. Then $F$ is closed in $M$.
Primitive of x fourth by Cosine of a x
:$\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$
Primitive of x sixth by Cosine of a x
:$\displaystyle \int x^6 \cos a x \rd x = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30 \sin a x} {a^3} x^4 - \frac {120 \cos a x} {a^4} x^3 + \frac {360 \sin a x} {a^5} x^2 + \frac {720 \cos a x} {a^6} x - \frac {720 \sin a x} {a^7} + C$
Fourier Series/Sixth Power of x over Minus Pi to Pi
:$\displaystyle x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$
Derivative of x to the a x
:$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$
Integer as Sum of Polygonal Numbers/Lemma 1
Let $n, m \in \N_{>0}$ such that $m \ge 3$. Let $n < 116 m$. Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$.
Integer as Sum of Polygonal Numbers/Lemma 3
Let $n, m, r \in \R_{>0}$. Suppose $\dfrac n m > 1$. Let $b \in \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$. Define: :$a = 2 \paren {\dfrac {n - b - r} m} + b = \paren {1 - \dfrac 2 m} b + 2 \paren {\dfrac {n - r} m}$ Then $a, b$ satisfy: :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$
Integer as Sum of Polygonal Numbers/Lemma 2
Let $n, m \in \R_{>0}$ such that $\dfrac n m \ge 1$. Define $I$ to be the open real interval: :$I = \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$ Then: :For $\dfrac n m \ge 116$, the length of $I$ is greater than $4$.
Integer as Sum of Three Odd Squares
Let $r$ be a positive integer. Then: :$r \equiv 3 \pmod 8$ : :$r$ is the sum of $3$ odd squares. {{explain|Could this be an  proof? Could also be named more precisely.</br>It is now. I referred to Integer as Sum of Three Squares when naming this. Could be classified as a corollary}}
Join Semilattice Ordered Subset Not Always Subsemilattice
Let $\struct {S, \circ}$ be a semilattices. Let $\preceq$ be the ordering on $S$ defined by: :$a \preceq b \iff \paren {a \circ b} = b$ Let $T$ be a subset of $S$. Let the ordered subset $\struct{T, \preceq \restriction_T}$ be a join semilattice. Let $\vee$ be the binary operation on $S$ defined by: :for all $a, b \in S$, $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$. Then: :$\struct{T, \vee}$ may not be a subsemilattice of $\struct {S, \circ}$.
Closure of Subset of Closed Set of Topological Space is Subset
Let $T$ = $\struct {S, \tau}$ be a topological space. Let $F$ be a closed set of $T$. Let $H \subseteq F$ be a subset of $F$. Let $H^-$ denote the closure of $H$. Then $H^- \subseteq F$.
Number of Partial Derivatives of Order n
Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \ldots, x_m$. There are $m^n$ partial derivatives of $u$ of order $n$.
Normed Vector Space is Open in Itself
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the set $X$ is an open set of $M$.
Normed Vector Space is Open in Itself/Proof 1
Let $M = \struct{X, \norm {\, \cdot \,}}$ be a normed vector space. Then the set $X$ is an open set of $M$.
Normed Vector Space is Open in Itself/Proof 2
Let $M = \struct{X, \norm {\, \cdot \,}}$ be a normed vector space. Then the set $X$ is an open set of $M$.
Empty Set is Closed in Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the empty set $\O$ is closed in $M$.
Empty Set is Open in Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the empty set $\O$ is an open set of $M$.
Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent
Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.  Then $\sequence {f_n}$ is uniformly Cauchy on $S$  $\sequence {f_n}$ converges uniformly on $S$.
Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition
Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Let $\sequence {f_n}$ be uniformly Cauchy on $S$.  Then $\sequence {f_n}$ is uniformly convergent on $S$.
Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition
Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Let $\sequence {f_n}$ be uniformly convergent on $S$. Then $\sequence {f_n}$ is uniformly Cauchy on $S$.
Open Ball of Point Inside Open Ball/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct{X, \norm {\, \cdot \,}}$. Let $y \in \map {B_\epsilon} x$. Then: : $\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$ That is, for every point in an open $\epsilon$-ball in a normed vector space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.
Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function
Let $S \subseteq \R$. Let $x \in S$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$ converging uniformly to $f : S \to \R$. Let $f_n$ be continuous at $x$ for all $n \in \N$. Then $f$ is continuous at $x$.
Open Ball is Open Set/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{>0}$. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon} x$ is an open set of $M$.
Finite Intersection of Open Sets of Normed Vector Space is Open
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $U_1, U_2, \ldots, U_n$ be open in $M$. Then $\displaystyle \bigcap_{i \mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open.
Union of Open Sets of Normed Vector Space is Open
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. The union of a set of open sets of $M$ is open in $M$.
Definite Integral of Limit of Uniformly Convergent Sequence of Integrable Functions
Let $a, b \in \R$ with $a < b$.  Let $\sequence {f_n}$ be a sequence of Riemann integrable real functions $\closedint a b \to \R$ converging uniformly to $f : \closedint a b \to \R$. Then $f$ is integrable and:  :$\displaystyle \int_a^b \map f x \rd x = \lim_{n \to \infty} \int_a^b \map {f_n} x \rd x$
Intersection of Closed Sets is Closed/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the intersection of an arbitrary number of closed sets of $M$ (either finitely or infinitely many) is itself closed.
Normed Vector Space is Closed in Itself
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then $X$ is closed in $M$.
Infinite Series of Functions is Uniformly Convergent iff Sequence of Partial Sums is Uniformly Cauchy
Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Then the infinite series:  :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly on $S$  for all $\varepsilon \in \R_{> 0}$ there exists $N \in \N$ such that: :$\displaystyle \size {\sum_{k \mathop = m + 1}^n \map {f_k} x} < \varepsilon$ for all $x \in S$ and $n > m > N$.
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Corollary
:$T_1 \cap T_2 = \overline {\overline T_1 \cup \overline T_2}$
LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma
Let $a, b, c \in \Z_{>0}$ be strictly positive integers. Then: :$\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } = \gcd \set {a, b, c}$
Events One of Which equals Intersection
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let $A$ and $B$ be such that: :$A \cap B = A$ Then whenever $A$ occurs, it is always the case that $B$ occurs as well.
Union of Event with Complement is Certainty
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$. Then: :$A \cup \overline A = \Omega$ where $\overline A$ is the complementary event to $A$. That is, $A \cup \overline A$ is a certainty.
Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 2
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}
Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 3
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}
Compact Sets in Fortissimo Space
A set in Fortissimo space is compact  it is finite.
Uniformly Convergent Series of Continuous Functions Converges to Continuous Function
Let $S \subseteq \R$. Let $x \in S$. Let $\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous at $x$ for all $n \in \N$. Let the infinite series:  :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ be uniformly convergent to a real function $f : S \to \R$. Then $f$ is continuous at $x$.
Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary
Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous for all $n \in \N$. Let the infinite series: :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ be uniformly convergent to a real function $f : S \to \R$. Then $f$ is continuous.
Euclidean Space is Banach Space/Proof 1
Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.
Euclidean Space is Banach Space/Proof 2
Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.
Monotone Function is of Bounded Variation
Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a monotone function. Then $f$ is of bounded variation.
Norm Equivalence is Equivalence
Let $X$ be a vector space. Let $\norm {\, \cdot \,}_a$ and $\norm {\, \cdot \,}_b$ be equivalent norms on $X$. Denote this relation by $\sim$:  :$\norm {\, \cdot \,}_a \sim \norm {\, \cdot \,}_b$. Then $\sim$ is an equivalence relation.
Continuous Non-Negative Real Function with Zero Integral is Zero Function
Let $a, b$ be real numbers with $a < b$.  Let $f : \closedint a b \to \R$ be a continuous function.  Let:  :$\map f x \ge 0$ for all $x \in \closedint a b$. Let:  :$\displaystyle \int_a^b \map f x \rd x = 0$ Then $\map f x = 0$ for all $x \in \closedint a b$.
Sequential Continuity is Equivalent to Continuity in the Reals/Sufficient Condition
Let $A \subseteq \R$ be a subset of the real numbers. Let $c \in A$. Let $f : A \to \R$ be a real function. Then if $f$ is continuous at $c$: :for each sequence $\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.
Sequential Continuity is Equivalent to Continuity in the Reals/Necessary Condition
Let $A \subseteq \R$ be a subset of the real numbers. Let $c \in A$. Let $f : A \to \R$ be a real function. Then if $f$ is continuous at $c$: :for each sequence $\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.
Metric Closure and Topological Closure of Subset are Equivalent
Let $M = \struct{A, d}$ be a metric space. Let $T = \struct{A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Then: :the metric closure of $H$ in $M$ equals the topological closure of $H$ in $T$
Set together with Condensation Points is not necessarily Closed
Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $\CC$ denote the set of condensation points of $H$. Then it is not necessarily the case that $H \cup \CC$ is a closed set of $T$.