hcju/proofwikips huggingface dataset:
Equivalence of Definitions of Transitive Closure (Relation Theory)/Finite Chain is Smallest
Let $S$ be a set or class. Let $\RR$ be a relation on $S$. Let $\RR^+$ be the transitive closure of $\RR$ by the finite chain definition. That is, for $x, y \in S$ let $x \mathrel {\RR^+} y$  for some natural number $n > 0$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and: :$\forall k \in \N_n: s_k \mathrel \RR s_{k+1}$ Then $\RR^+$ is transitive and if $\QQ$ is a transitive relation on $S$ such that $\RR \subseteq \QQ$ then $\RR \subseteq \QQ$.
Order-Preserving Bijection on Wosets is Order Isomorphism
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets. Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving: :$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$ Then: :$\forall x, y \in S: \map \phi x \preceq_2 \map \phi y \implies x \preceq_1 y$ That is, $\phi: S \to T$ is an order isomorphism.
Inverse Image of Set under Set-Like Relation is Set
Let $A$ be a class. Let $\RR$ be a set-like endorelation on $A$. Let $B \subseteq A$ be a set. Then $\map {\RR^{-1} } B$, the inverse image of $B$ under $\RR$, is also a set.
Reciprocal of Holomorphic Function
Let $f: \C \to \C$ be a complex function. Let $U \subseteq \C$ be an open set such that $f$ has no zeros in $U$. Suppose further that $f$ is holomorphic in $U$. Then the complex function :$\dfrac 1 {f_{\restriction U} } : U \to \C$ is holomorphic.
Transitive Closure of Set-Like Relation is Set-Like
Let $A$ be a class. Let $\RR$ be a set-like endorelation on $A$. Let $\RR^+$ be the transitive closure of $\RR$. Then $\RR^+$ is also a set-like relation.
Minimal WRT Restriction
Let $A$ be a set or class. Let $\mathcal R$ be a relation on $A$. Let $B$ be a subset or subclass of $A$. Let $\mathcal R'$ be the restriction of $\mathcal R$ to $B$. Let $m \in B$. Then: :$m$ is $\mathcal R$-minimal in $B$ : :$m$ is $\mathcal R'$-minimal in $B$.
Intersection of Ordinals is Ordinal
Let $A$ be a non-empty class of ordinals. Then $\bigcap A$ is an ordinal.
Meet with Complement is Bottom
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 2. Then: :$\exists \bot \in S: \forall a \in S: a \wedge \neg a = \bot$ where $\wedge$ denotes the meet operation in $S$. This element $\bot$ is unique for any given $S$, and is named '''bottom'''.
Join with Complement is Top
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 2. Then: :$\exists \top \in S: \forall a \in S: a \vee \neg a = \top$ where $\wedge$ denotes the meet operation in $S$. This element $\top$ is unique for any given $S$, and is named '''top'''.
Ordering is Equivalent to Subset Relation/Lemma
Let $\struct {S, \preceq}$ be an ordered set. Then: : $\forall a_1, a_2 \in S: \paren ({a_1 \preceq a_2 \implies {a_1}^\preceq \subseteq {a_2}^\preceq}$ where ${a_1}^\preceq$ denotes the lower closure of $a_1$.
Smallest Element WRT Restricted Ordering
Let $S$ be a set or class. Let $\preceq$ be an ordering on $S$. Let $T$ be a subset or subclass of $S$. Let $\preceq'$ be the restriction of $\preceq$ to $T$. Let $m \in T$. Then $m$ is the $\preceq$-smallest element of $T$ iff $m$ is the $\preceq'$-smallest element of $T$.
Restriction of Well-Founded Ordering
Let $S$ be a set or class. Let $T$ be a subset or subclass of $S$. Let $\preceq$ be a well-founded ordering of $A$. Let $\preceq'$ be the restriction of $\preceq$ to $T$. Then $\preceq'$ is a well-founded ordering of $T$.
Restriction of Well-Ordering is Well-Ordering
Let $S$ be a set or class. Let $\preceq$ be a well-ordering of $S$. Let $T$ be a subset or subclass of $S$. Let $\preceq'$ be the restriction of $\preceq$ to $T$. Then $\preceq'$ well-orders $T$.
Ordering is Equivalent to Subset Relation/Proof 2
{{:Ordering is Equivalent to Subset Relation}} Specifically: Let :$\mathbb S := \set {a^\preceq: a \in S}$ where $a^\preceq$ is the lower closure of $a$. That is: :$a^\preceq := \set {b \in S: b \preceq a}$ Let the mapping $\phi: S \to \mathbb S$ be defined as: :$\map \phi a = a^\preceq$ Then $\phi$ is an order isomorphism from $\struct {S, \preceq}$ to $\struct {\mathbb S, \subseteq}$.
Event Space contains Sample Space
:$\Omega \in \Sigma$
Event Space contains Empty Set
:$\O \in \Sigma$
Power Set of Sample Space is Event Space
Let $\EE$ be an experiment whose sample space is $\Omega$. Let $\powerset \Omega$ be the power set of $\Omega$. Then $\powerset \Omega$ is an event space of $\EE$.
Event Space from Single Subset of Sample Space
Let $\EE$ be an experiment whose sample space is $\Omega$. Let $\O \subsetneqq A \subsetneqq \Omega$. Then $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ is an event space of $\EE$.
Intersection of Events is Event
:$A, B \in \Sigma \implies A \cap B \in \Sigma$
Set Difference of Events is Event
:$A, B \in \Sigma \implies A \setminus B \in \Sigma$
Symmetric Difference of Events is Event
:$A, B \in \Sigma \implies A \ast B \in \Sigma$
Characterization of Affine Transformations
Let $\mathcal E$ and $\mathcal F$ be affine spaces over a field $k$. Let $\mathcal L: \mathcal E \to \mathcal F$ be a mapping. Then $\mathcal L$ is an affine transformation  for all points $p, q \in \mathcal E$ and all $\lambda \in k$: :$\mathcal L \left({\lambda p + \left({1 - \lambda}\right) q}\right) = \lambda \mathcal L \left({p}\right) + \left({1 - \lambda}\right) \mathcal L \left({q}\right)$ where $\lambda p + \left({1 - \lambda}\right) q$ and $\lambda \mathcal L \left({p}\right) + \left({1 - \lambda}\right) \mathcal L \left({q}\right)$ denote barycenters.
Probability of Empty Event is Zero
:$\map \Pr \O = 0$
Ordinal Membership is Asymmetric
Let $m$ and $n$ be ordinals. Then it is not the case that $m \in n$ and $n \in m$.
Intersection of Ordinals is Smallest
Let $A$ be a non-empty set or class of ordinals. Let $m = \bigcap A$ be the intersection of all the elements of $A$. Then $m$ is the smallest element of $A$.
Ordinal Class is Strongly Well-Ordered by Subset
Let $\On$ be the class of all ordinals. Then the restriction of the subset relation, $\subseteq$, to $\On$ is a strong well-ordering. That is: :$\subseteq$ is an ordering on $\On$. :If $A$ is a non-empty subclass of $\On$, then $A$ has a $\subseteq$-smallest element.
Subset is Compatible with Ordinal Successor
Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x \in y$. Then: :$x^+ \in y^+$
Successor is Less than Successor/Sufficient Condition/Proof 1
Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x^+ \in y^+$. Then: : $x \in y$
Successor is Less than Successor/Sufficient Condition/Proof 2
Let $x$ and $y$ be ordinals and let $x^+$ denote the successor set of $x$. Let $x^+ \in y^+$. Then: : $x \in y$
Successor is Less than Successor/Sufficient Condition
== Proof 1 == {{:Successor is Less than Successor/Sufficient Condition/Proof 1}}
Probability of Event not Occurring
:$\forall A \in \Sigma: \map \Pr {\Omega \setminus A} = 1 - \map \Pr A$
Delta-Algebra is Sigma-Algebra
Every $\delta$-algebra is a $\sigma$-algebra.
Countable Intersection of Events is Event
:$\quad A_1, A_2, \ldots \in \Sigma \implies \ds \bigcap_{i \mathop = 1}^\infty A_i \in \Sigma$
Subset Relation is Ordering/General Result
Let $\mathbb S$ be a set of sets or class. Then $\subseteq$ is an ordering on $\mathbb S$. In other words, let $\left({\mathbb S, \subseteq}\right)$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$. Then $\left({\mathbb S, \subseteq}\right)$ is an ordered set.
Noetherian Domain is Factorization Domain
Let $R$ be a noetherian integral domain. Then $R$ is a factorization domain.
Probability Measure is Monotone
Let $A, B \in \Sigma$ such that $A \subseteq B$. Then: :$\map \Pr A \le \map \Pr B$
Transfinite Induction/Principle 1/Proof 2
Let $\On$ denote the class of all ordinals. Let $A$ denote a class. Suppose that: :For each element $x$ of $\On$, if $\forall y \in \On: \paren {y < x \implies y \in A}$ then $x$ is an element of $A$. Then $\On \subseteq A$.
Immediate Successor is Unique in Toset
Let $(S, \preceq)$ be a totally ordered set. Let $x, y \in S$. Suppose that $y$ is an immediate successor of $x$. Then $y$ is the ''only'' immediate successor of $x$.
Mapping from Set to Ordinal Class is Bounded Above
Let $x$ be a set. Let $\operatorname{On}$ be the class of all ordinals. Let $f: x \to \operatorname{On}$ be a mapping. Then $f$ has an upper bound.
Equivalence of Definitions of Independent Events
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$. {{TFAE|def = Independent Events}}
Union of Subset of Ordinals is Ordinal/Corollary
Let $y$ be a set. Let $\On$ be the class of all ordinals. Let $F: y \to \On$ be a mapping. Then: :$\displaystyle \bigcup \map F y \in \On$ where $\map F y$ is the image of $y$ under $F$.
Independent Events are Independent of Complement/General Result
Let $A_1, A_2, \ldots, A_m$ be events in a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $A_1, A_2, \ldots, A_m$ are independent  $\Omega \setminus A_1, \Omega \setminus A_2, \ldots, \Omega \setminus A_m$ are also independent.
Probability of Independent Events Not Happening/Corollary
Let $A$ be an event in an event space of an experiment $\EE$ whose probability space is $\struct {\Omega, \Sigma, \Pr}$. Let $\map \Pr A = p$. Suppose that the nature of $\EE$ is that its outcome is independent of previous trials of $\EE$. Then the probability that $A$ does not occur during the course of $m$ trials of $\EE$ is $\paren {1 - p}^m$.
Probability of Limit of Sequence of Events/Increasing
Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N}$ be an increasing sequence of events. Let $\displaystyle A = \bigcup_{i \mathop \in \N} A_i$ be the limit of $\left \langle{A_n}\right \rangle_{n \mathop \in \N}$. Then: :$\displaystyle \Pr \left({A}\right) = \lim_{n \to \infty} \Pr \left({A_n}\right)$
Probability of Limit of Sequence of Events/Decreasing
Let $\left \langle{B_n}\right \rangle_{n \mathop \in \N}$ be a decreasing sequence of events. Let $\displaystyle B = \bigcap_{i \mathop \in \N} B_i$ be the limit of $\left \langle{B_n}\right \rangle_{n \mathop \in \N}$. Then: :$\displaystyle \Pr \left({B}\right) = \lim_{n \to \infty} \Pr \left({B_n}\right)$
Sum of Discrete Random Variables
Let $U: \Omega \to \R$ be defined as: :$\forall \omega \in \Omega: \map U \omega = \map X \omega + \map Y \omega$ Then $U$ is also a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Product of Discrete Random Variables
Let $V: \Omega \to \R$ be defined as: :$\forall \omega \in \Omega: V \left({\omega}\right) = X \left({\omega}\right) Y \left({\omega}\right)$ Then $V$ is also a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.
Bernoulli Process as Geometric Distribution/Shifted
Let $\left \langle{Y_i}\right \rangle$ be a Bernoulli process with parameter $p$. Let $\mathcal E$ be the experiment which consists of performing the Bernoulli trial $Y_i$ as many times as it takes to achieve a success, and then stop. Let $k$ be the number of Bernoulli trials to achieve a success. Then $k$ is modelled by a shifted geometric distribution with parameter $p$.
Geometric Distribution Gives Rise to Probability Mass Function/Shifted
Let $Y$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Let $Y$ have the shifted geometric distribution with parameter $p$ (where $0 < p < 1$). Then $Y$ gives rise to a probability mass function.
Negative Binomial Distribution Gives Rise to Probability Mass Function
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Negative Binomial Distribution Gives Rise to Probability Mass Function/First Form
Let $X$ have the negative binomial distribution (first form) with parameters $n$ and $p$ ($0 < p < 1$). Then $X$ gives rise to a probability mass function.
Bernoulli Process as Negative Binomial Distribution/First Form
Let $\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a total of $n$ failures have been encountered. Let $X$ be the discrete random variable defining the number of successes before $n$ failures have been encountered. Then $X$ is modeled by a negative binomial distribution of the first form.
Bernoulli Process as Negative Binomial Distribution/Second Form
Let $\EE$ be the experiment which consists of performing the Bernoulli trial $X_i$ as many times as it takes to achieve a total of $n$ successes, and then stops. Let $Y$ be the discrete random variable defining the number of trials before $n$ successes have been achieved. Then $X$ is modeled by a negative binomial distribution of the second form.
Negative Binomial Distribution as Generalized Geometric Distribution/Second Form
The second form of the negative binomial distribution is a generalization of the shifted geometric distribution: Let $\sequence {Y_i}$ be a Bernoulli process with parameter $p$. Let $\FF$ be the experiment which consists of: :Perform the Bernoulli trial $Y_i$ as many times as it takes to achieve $n$ successes, and then stop. Let $k$ be the number of Bernoulli trials that need to be taken in order to achieve up to (and including) the $n$th success. Let $\FF'$ be the experiment which consists of: :Perform the Bernoulli trial $Y_i$ until '''one''' success is achieved, and then stop. Then $k$ is modelled by the experiment: :Perform experiment $\FF'$ until $n$ failures occur, and then stop.
Derivative of Geometric Sequence/Corollary
:$\displaystyle \sum_{n \mathop \ge 1} n \paren {n + 1} x^{n - 1} = \frac 2 {\paren {1 - x}^3}$
Equivalence of Definitions of Countable Set
Let $S$ be a set. {{TFAE|def = Countable Set}}
Infinite Set of Natural Numbers is Countably Infinite
Let $\N$ be the set of natural numbers. Let $S$ be an infinite subset of $\N$. Then $S$ is countably infinite. That is, there is a bijection $f: \N \to S$.
Generating Function for Natural Numbers
Let $\sequence {a_n}$ be the sequence defined as: : $\forall n \in \N_{> 0}: a_n = n - 1$ That is: :$\sequence {a_n} = 0, 1, 2, 3, 4, \ldots$ Then the generating function for $\sequence {a_n}$ is given as: :$G \paren z = \dfrac 1 {\paren {1 - z}^2}$
Injection has Surjective Left Inverse Mapping
Let $S$ and $T$ be sets such that $S \ne \O$. Let $f: S \to T$ be a injection. Then there exists a surjection $g: T \to S$ such that: :$g \circ f = I_S$
Many-to-One Relation Extends to Mapping
Let $S$ and $T$ be sets. Let $T$ be non-empty. Let $\mathcal R \subset S \times T$ be a many-to-one relation. Then there exists a mapping $f: S \to T$ such that $\mathcal R \subseteq f$.
Probability Generating Function as Expectation
Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$. Let $p_X$ be the probability mass function for $X$. Let $\Pi_X \left({s}\right)$ be the probability generating function for $X$. Then: :$\Pi_X \left({s}\right) = E \left({s^X}\right)$ where $E \left({s^X}\right)$ denotes the expectation of $s^X$.
Probability Generating Function of Negative Binomial Distribution/First Form
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$ where $q = 1 - p$.
Probability Generating Function of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the p.g.f. of $X$ is: :$\displaystyle \Pi_X \left({s}\right) = \left({\frac {ps} {1 - qs}}\right)^n$ where $q = 1 - p$.
Probability Generating Function of Scalar Multiple of Random Variable
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Let $k \in \Z_{\ge 0}$ be a positive integer. Let $Y$ be a discrete random variable such that $Y = m X$. Then :$\Pi_Y \left({s}\right) = \Pi_X \left({s^m}\right)$. where $\Pi_Y \left({s}\right)$ is the probability generating function of $Y$.
Derivatives of Probability Generating Function at One
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Then the $n$th derivative of $\Pi_X \left({s}\right)$ at $s = 1$ is given by: :$\dfrac {\mathrm d^n} {\mathrm d s^n} \Pi_X \left({1}\right) = E \left({X \left({X - 1}\right) \cdots \left({X - n + 1}\right)}\right)$ for $n = 1, 2, \ldots$
Differentiation of Power Series/Corollary
The value of $\displaystyle \frac {\mathrm d^n}{\mathrm d x^n} \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m$ at $x = \xi$ is: :$\displaystyle \left.{\frac {\mathrm d^n}{\mathrm d x^n} \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m}\right|_{x = \xi} = a_n n!$
Non-Equivalence of Proposition and Negation/Formulation 1
:$p \implies \neg p, \neg p \implies p \vdash \bot$
Non-Equivalence of Proposition and Negation/Formulation 2
:$\vdash \neg \left({p \iff \neg p}\right)$
No Injection from Power Set to Set/Lemma
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\powerset S$.
No Injection from Power Set to Set
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then there is no injection from $\powerset S$ into $S$.
Inverse Image Mapping of Injection is Surjection
Let $S$ and $T$ be sets. Let $f: S \to T$ be a injection. Let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$. Then $f^\gets$ is a surjection.
Expectation of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the expectation of $X$ is given by: :$E \left({X}\right) = \dfrac n p$
Derivatives of PGF of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = ...$
First Derivative of PGF of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the first derivative of the PGF of $X$ {{WRT|Differentiation}} $s$ is: :$\dfrac \d {\d s} \map {\Pi_X} s = n p \paren {\dfrac {\paren {p s}^{n - 1} } {\paren {1 - q s}^{n + 1} } }$
Nth Derivative of Reciprocal of Mth Power/Corollary
The $n$th derivative of $\dfrac 1 x$ {{WRT|Differentiation}} $x$ is: :$\dfrac {\d^n} {\d x^n} \dfrac 1 x = \dfrac {\paren {-1}^n n!} {x^{n + 1} }$ where $n!$ denotes $n$ factorial.
First Derivative of PGF of Negative Binomial Distribution/First Form
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the first derivative of the PGF of $X$ w.r.t. $s$ is: :$\dfrac \d {\d s} \map {\Pi_X} s = \dfrac {n p} q \paren {\dfrac q {1 - p s} }^{n + 1}$
Expectation of Negative Binomial Distribution/First Form
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$. Then the expectation of $X$ is given by: :$E \left({X}\right) = \dfrac {n p} q$
Closure Condition for Hausdorff Space
Let $\struct {X, \tau}$ be a topological space. Then $\struct {X, \tau}$ is a Hausdorff space : :For all $x, y \in X$ such that $x \ne y$, there exists an open set $U$ such that $x \in U$ and $y \notin U^-$, where $U^-$ is the closure of $U$.
Second Derivative of PGF of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the second derivative of the PGF of $X$ w.r.t. $s$ is: :$\dfrac {\mathrm d^2} {\mathrm d s^2} \Pi_X \left({s}\right) = \left({\dfrac {ps} {1 - qs} }\right)^{n+2} \left({\dfrac {n \left({n-1}\right) + 2 n q s} {\left({p s^2}\right)^2} }\right)$
Derivatives of PGF of Negative Binomial Distribution/First Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \dfrac {n^{\overline k} p^k} {q^k} \left({\dfrac q {1 - p s} }\right)^{n + k}$ where: : $n^{\overline k}$ is the rising factorial: $n^{\overline k} = n \left({n + 1}\right) \left({n + 2}\right) \cdots \left({n + k - 1}\right)$ : $q = 1 - p$
Variance of Negative Binomial Distribution/Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$. Then the variance of $X$ is given by: :$\displaystyle \operatorname{var} \left({X}\right) = \frac {n q} {p^2}$
PGF of Sum of Independent Discrete Random Variables/General Result
Let: :$Z = X_1 + X_2 + \cdots + X_n$ where each of $X_1, X_2, \ldots, X_n$ are independent discrete random variables with PGFs $\map {\Pi_{X_1} } s, \map {\Pi_{X_2} } s, \ldots, \map {\Pi_{X_n} } s$. Then: :$\displaystyle \map {\Pi_Z} s = \prod_{j \mathop = 1}^n \map {\Pi_{X_j} } s$
Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma
Let $\left({S, \tau}\right)$ be a Hausdorff space. Let $C$ be a compact subspace of $S$. Let $x \in S \setminus C$. Then there exist open sets $U$ and $V$ such that $x \in U$, $C \subseteq V$, and $U \cap V = \varnothing$.
Sum of Independent Binomial Random Variables
Let $X$ and $Y$ be discrete random variables with a binomial distribution: :$X \sim \operatorname{B} \left({m, p}\right)$ and :$Y \sim \operatorname{B} \left({n, p}\right)$ Let $X$ and $Y$ be independent. Then their sum $Z = X + Y$ is distributed as: :$Z \sim \operatorname{B} \left({\left({m + n}\right), p}\right)$
Field has Algebraic Closure
Every field has an algebraic closure.
Point Finite Set of Open Sets in Separable Space is Countable
Let $\struct {X, \tau}$ be a separable space. Let $\mathcal F$ be a point finite set of open sets of $X$. Then $\mathcal F$ is countable.
Countable Union of Finite Sets is Countable
The following statements are equivalent in $\mathrm{ZF}^-$: {{explain|$\mathrm{ZF}^-$}} :The Axiom of Countable Choice for Finite Sets holds. :The union of any countable set of finite sets is countable.
Reverse Triangle Inequality/Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Then: :$\forall x, y \in X: \norm {x - y} \ge \size {\norm x - \norm y}$
Equivalence of Forms of Axiom of Countable Choice
The following forms of the Axiom of Countable Choice are equivalent in $\mathrm{ZF}^-$: {{explain|$\mathrm{ZF}^-$}}
GCD from Generator of Ideal
Let $m, n \in \Z$, with either $m \ne 0$ or $n \ne 0$. Let $I = \ideal {m, n}$ be the ideal generated by $m$ and $n$. Let $d$ be a non-negative generator for the principal ideal $I$. Then: :$\gcd \set {m, n} =  d$ where $\gcd \set {m, n}$ denotes the greatest common divisor of $m$ and $n$.
Finite Union of Finite Sets is Finite
Let $S$ be a finite set of finite sets. Then the union of $S$ is finite.
Open Set Characterization of Denseness
Let $\left({X, \tau}\right)$ be a topological space. Let $S \subseteq X$. Then $S$ is (everywhere) dense in $X$  every non-empty ($\tau$-)open set contains an element of $S$.
Union of Set of Sets when a Set Intersects All
Let $F$ be a set of sets. Let $S$ be a set or class. Suppose that: :$\forall A \in F: A \cap S \ne \O$ Then: :$\ds F = \bigcup_{x \mathop \in S} \set {A \in F: x \in A}$
Order of Group of Units of Integers Modulo m
Let $n \in \Z_{\ge 0}$ be an integer. Let $\struct {\Z / n \Z, +, \cdot}$ be the ring of integers modulo $n$. Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the group of units of this ring. Then: :$\order U = \map \phi n$ where $\phi$ denotes the Euler $\phi$-function.