hcju/proofwikips huggingface dataset:
Matrix Multiplication Interpretation of Relation Composition
Let $A$, $B$ and $C$ be finite non-empty sets that are initial segments of $\N_{\ne 0}$. Let $\mathcal R \subseteq B \times A$ and $\mathcal S \subseteq C \times B$ be relations. Let $\mathbf R$ and $\mathbf S$ be matrices which we define as follows: :$\left[{r}\right]_{i j} = \begin{cases} T & : (i, j) \in \mathcal R \\ F & : (i, j) \notin \mathcal R\\ \end{cases}$ :$\left[{s}\right]_{i j} = \begin{cases} T & : (i, j) \in \mathcal S \\ F & : (i, j) \notin \mathcal S\\ \end{cases}$ Then we can interpret the matrix product $\mathbf R \mathbf S$ as the composition $\mathcal S \circ \mathcal R$. To do so we temporarily consider $\left({\left\{ {T, F}\right\}, \land, \lor}\right)$ to be our "ring" on which we are basing matrix multiplication. Then: :$\left[{r s}\right]_{i j} = T \iff (i, j) \in \mathcal S \circ \mathcal R$
Set of Subsets is Cover iff Set of Complements is Free
Let $S$ be a set. Let $\mathcal C$ be a set of sets. Then $\mathcal C$ is a cover for $S$  $\set {\relcomp S X: X \in \mathcal C}$ is free.
Metric Space is Perfectly Normal
Let $M = \struct {A, d}$ be a metric space. Then $M$ is a perfectly normal space.
Tutte's Wheel Theorem
Every 3-connected graph can be obtained by the following procedure: * Start with $G_0 := K_4$ * Given $G_i$ pick a vertex $v$ * Split into $v'$ and $v''$ and add edge $\{v',v''\}$ This procedure directly follows from the theorem:  :A graph $G$ is 3-connected ('''A''') iff there exists a sequence $G_0, G_1, . . . , G_n$ of graphs with the following properties ('''B'''):  :* $G_0 = K_4$ and $G_n = G$; :* $G_{i+1}$ has an edge $e = xy$ with $deg \left({x}\right), deg \left({y}\right) \geq 3$ and $G_i = G_{i+1} \thinspace / \thinspace e$ for every $i < n$.
G-Delta Sets in Indiscrete Topology
$H$ is a $G_\delta$ ($G$-delta) set of $T$  either $H = S$ or $H = \O$.
Subset of Indiscrete Space is Sequentially Compact
$H$ is sequentially compact in $T$.
Infinite Particular Point Space is not Metacompact
Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space. Then $T$ is not metacompact.
Condition for Closed Extension Space to be T4 Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then: :$T^*_p$ is a $T_4$ space  $T$ is a $T_4$ space vacuously and $T^*_p$ in this case is also a $T_4$ space vacuously.
Condition for Closed Extension Space to be T5 Space
Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then: :$T^*_p$ is a $T_5$ space  $T$ is a $T_5$ space vacuously and $T^*_p$ in this case is also a $T_5$ space vacuously.
Separation Properties of Alexandroff Extension of Rational Number Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ satisfies no Tychonoff separation axioms higher than a $T_1$ (Fréchet) space.
Hilbert Sequence Space is not Locally Compact Hausdorff Space
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is not a locally compact Hausdorff space.
Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point
Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Let $A \subseteq S$ be a connected space. Let $p \in A$ be a point of $A$ which is not an endpoint of $A$. Then $p$ is a cut point of $A$.
Sum of Infinite Series of Product of Power and Sine
Let $r \in \R$ such that $\size r < 1$. Let $x \in \R$ such that $x \ne 2 m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^\infty r^k \sin k x       | r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots       | c =  }} {{eqn | r = \dfrac {r \sin x} {1 - 2 r \cos x + r^2}       | c =  }} {{end-eqn}}
Coefficients of Cosine Terms in Convergent Trigonometric Series
Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\openint \alpha {\alpha + 2 \pi}$: :$\map f x = \dfrac {a_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \paren {a_m \cos m x + b_m \sin m x}$ Then: :$\forall n \in \Z_{\ge 0}: a_n = \dfrac 1 \pi \displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x$
De Morgan's Laws (Logic)/Conjunction/Formulation 2/Proof 2
: $\vdash \left({p \land q}\right) \iff \left({\neg \left({\neg p \lor \neg q}\right)}\right)$
Derivation of Fourier Series over General Range
Let $\alpha \in \R$ be a real number. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $f: \R \to \R$ be a function such that $\displaystyle \int_{\mathop \to \alpha}^{\mathop \to \alpha + 2 \lambda} \map f x \rd x$ converges absolutely. Let: :$\displaystyle f \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$ The '''Fourier coefficients''' for $f$ are calculated by: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \lambda \int_{\mathop \to \alpha}^{\mathop \to \alpha + 2 \lambda} \map f x \cos \frac {n \pi x} \lambda \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \lambda \int_{\mathop \to \alpha}^{\mathop \to \alpha + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x }} {{end-eqn}}
Half-Range Fourier Sine Series/Cosine over 0 to Pi
:500pxrightthumb$\map f x$ and its $7$th approximation On the interval $\openint 0 \pi$: {{begin-eqn}} {{eqn | l = \cos x       | r = \frac 8 \pi \sum_{m \mathop = 1}^\infty \frac {m \sin 2 m x} {4 m^2 - 1}       | c =  }} {{eqn | r = \frac 8 \pi \paren {\frac {\sin 2 x} {1 \times 3} + \frac {2 \sin 4 x} {3 \times 5} + \frac {3 \sin 6 x} {5 \times 7} + \cdots}       | c =  }} {{end-eqn}}
Properties of Ordered Group/OG2/Proof 1
Let $\left({G, \circ, \preceq}\right)$ be an ordered group with identity $e$. Let $\prec$ be the reflexive reduction of $\preceq$. Let $x, y \in G$. Then the following equivalences hold: :$(\operatorname{OG}2.1):\quad x \preceq y \iff e \preceq y \circ x^{-1}$ :$(\operatorname{OG}2.2):\quad x \preceq y \iff e \preceq x^{-1} \circ y$ :$(\operatorname{OG}2.3):\quad x \preceq y \iff x \circ y^{-1} \preceq e$ :$(\operatorname{OG}2.4):\quad x \preceq y \iff y^{-1} \circ x \preceq e$ :$(\operatorname{OG}2.1'):\quad x \prec y \iff e \prec y \circ x^{-1}$ :$(\operatorname{OG}2.2'):\quad x \prec y \iff e \prec x^{-1} \circ y$ :$(\operatorname{OG}2.3'):\quad x \prec y \iff x \circ y^{-1} \prec e$ :$(\operatorname{OG}2.4'):\quad x \prec y \iff y^{-1} \circ x \prec e$
Series Expansion for Pi over Root 2
:$\displaystyle \frac \pi {\sqrt 2} = \sum_{r \mathop = 1}^\infty \paren {-1}^{r - 1} \frac {r - \frac 1 2} {r^2 - r + \frac 3 {16} }$
Leading Coefficient of Product of Polynomials over Integral Domain
Let $R$ be an integral domain. Let $f, g \in R \sqbrk x$ be polynomials. Let $c$ and $d$ be their leading coefficients. Then $f g$ has leading coefficient $c d$.
Ordered Set is Upper Set in Itself
Let $(S, \preceq)$ be an ordered set. Then $S$ is an upper set in $S$.
Ideal Contained in Finite Union of Prime Ideals
Let $A$ be a commutative ring with unity. Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be prime ideals. Let $\mathfrak a \subseteq \ds \bigcup_{i \mathop = 1}^n \mathfrak p_i$ be an ideal contained in their union. Then $\mathfrak a \subseteq \mathfrak p_i$ for some $i \in \{1, \ldots, n\}$.
Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence
Let $B$ be a algebra over $A$ that is unital, associative and commutative. Let $(C, f)$ be a ring under $A$.  #$C$ is the underlying ring of $B$ and $f : A \to C$ is the canonical mapping to the unital algebra $B$. #$B$ is the algebra defined by $f$.
Cayley-Hamilton Theorem/Matrices
Let $A$ be a commutative ring with unity. Let $\mathbf N = \left({a_{ij} }\right)$ be an $n \times n$ matrix with entries in $A$. Let $\mathbf I_n$ denote the $n \times n$ unit matrix. Let $p_N \left({x}\right)$ be the determinant $\det \left({x \cdot \mathbf I_n - \mathbf N}\right)$. Then: : $p_N \left({N}\right) = \mathbf 0$ as an $n \times n$ zero matrix. That is: :$ N^n + b_{n-1} N^{n-1} + \cdots + b_1 N + b_0 = \mathbf 0$ where the $b_i$ are the coefficients of $p_N \left({x}\right)$.
Subalgebra of Algebraic Field Extension is Field
Let $E / F$ be an algebraic field extension. Let $A \subseteq E$ be a unital subalgebra over $F$. Then $A$ is a field.
Modulus of Gamma Function of One Half plus Imaginary Number
Let $t \in \R$ be a real number. Then:  :$\cmod {\map \Gamma {\dfrac 1 2 + i t} } = \sqrt {\pi \map \sech {\pi t} }$ where: :$\Gamma$ is the Gamma function :$\sech$ is the hyperbolic secant function.
Factor Principles/Conjunction on Right/Formulation 1/Proof 1
: $p \implies q \vdash \left({p \land r}\right) \implies \left ({q \land r}\right)$
Equivalence of Definitions of Reflexive Closure
{{TFAE|def = Reflexive Closure}} Let $\RR$ be a relation on a set $S$.
Definite Integral of Constant Multiple of Real Function
Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. {{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}} Let $c \in \R$ be a real number. Then: :$\displaystyle \int_a^b c \map f x \rd x = c \int_a^b \map f x \rd x$
Mean Value Theorem for Integrals/Generalization
Let $f$ and $g$ be continuous real functions on the closed interval $\closedint a b$ such that: :$\forall x \in \closedint a b: \map g x \ge 0$ Then there exists a real number $k \in \closedint a b$ such that: :$\displaystyle \int_a^b \map f x \map g x \rd x = \map f k \int_a^b \map g x \rd x$
Barycenter Exists and is Well Defined
Let $\mathcal E$ be an affine space over a field $k$. Let $p_1, \ldots, p_n \in \mathcal E$ be points. Let $\lambda_1, \ldots, \lambda_n \in k$ such that $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$. Then the barycentre of $p_1, \ldots, p_n$ with weights $\lambda_1, \ldots, \lambda_n$ exists and is unique.
Definite Integral to Infinity of Power of x over Power of x plus Power of a
:$\displaystyle \int_0^\infty \dfrac {x^m \rd x} {x^n + a^n} = \frac {\pi a^{m + 1 - n} } {n \map \sin {\paren {m + 1} \frac \pi n} }$ for $0 < m + 1 < n$.
Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared
:$\displaystyle \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} } = \frac \pi 2$ for $a > 0$.
Definite Integral from 0 to Pi of Sine of m x by Sine of n x
Let $m, n \in \Z$ be integers. Then: :$\displaystyle \int_0^\pi \sin m x \sin n x \rd x = \begin{cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end{cases}$ That is: :$\displaystyle \int_0^\pi \sin m x \sin n x \rd x = \dfrac \pi 2 \delta_{m n}$ where $\delta_{m n}$ is the Kronecker delta.
Equivalence of Definitions of Strict Ordering
Let $S$ be a set. Let $\RR$ be a relation on $S$. {{TFAE|def = Strict Ordering}}
Subband of Induced Operation is Set of Subbands
Let $\left({S, \circ}\right)$ be a band. Let $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of: : the power set $\mathcal P \left({S}\right)$ of $S$ and : the operation $\circ_\mathcal P$ induced on $\mathcal P \left({S}\right)$ by $\circ$. Let $T \subseteq \mathcal P \left({S}\right)$. Let $\left({T, \circ_\mathcal P}\right)$ be a subband of $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$. Then every element of $T$ is a subband of $\left({S, \circ}\right)$.
Equivalence of Definitions of Closed Element
Let $\struct {S, \preceq}$ be an ordered set. Let $\cl$ be a closure operator on $S$. Let $x \in S$. {{TFAE|def = Closed Element}}
Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign
{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^n} {\paren {2 n + 1}^3}       | r = \frac 1 {1^3} - \frac 1 {3^3} + \frac 1 {5^3} - \frac 1 {7^3} + \cdots       | c =  }} {{eqn | r = \frac {\pi^3} {32}       | c =  }} {{end-eqn}}
Closed Element of Composite Closure Operator
Let $\left({S, \preceq}\right)$ be an ordered set. Let $f, g: S \to S$ be closure operators. Let $h = f \circ g$, where $\circ$ represents composition. Suppose that $h$ is also a closure operator. Then an element $x \in S$ is closed with respect to $h$ iff it is closed with respect to $f$ and with respect to $g$.
Power Series Expansion of Reciprocal of Square Root of 1 + x
Let $x \in \R$ such that $-1 < x \le 1$. Then: {{begin-eqn}} {{eqn | l = \dfrac 1 {\sqrt {1 + x} }       | r = \sum_{k \mathop = 0}^\infty \left({-1}\right)^k \frac {\left({2 k}\right)!} {\left({2^k k!}\right)^2} x^k       | c =  }} {{eqn | r = 1 - \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots       | c =  }} {{end-eqn}}
Inverse Image under Embedding of Image under Relation of Image of Point
Let $S$ and $T$ be sets. Let $\mathcal R_S$ and $\mathcal R_t$ be relations on $S$ and $T$, respectively. Let $\phi: S \to T$ be a mapping with the property that: : $\forall p, q \in S: \left({ p \mathrel{\mathcal R_S} q \iff \phi(p) \mathrel{\mathcal R_T} \phi(q) }\right)$ Then for each $p \in S$: : $\mathcal R_S (p) = \phi^{-1}\left({\mathcal R_T \left({ \phi(p) }\right) }\right)$
Power Series Expansion of Cube Root of 1 + x
Let $x \in \R$ such that $-1 < x \le 1$. Then: :$\dfrac 1 {\sqrt [3] {1 + x} } = 1 + \dfrac 1 3 x - \dfrac 2 {3 \times 6} x^2 + \dfrac {2 \times 5} {3 \times 6 \times 9} x^3 - \cdots$
Power Series Expansion for Real Arcsecant Function
The arcsecant function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \operatorname {arcsec} x       | r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2 \left({2 n + 1}\right) x^{2 n + 1} }       | c =  }} {{eqn | r = \frac \pi 2 - \left({\frac 1 x + \frac 1 2 \frac 1 {3 x^3} + \frac {1 \times 3} {2 \times 4} \frac 1 {5 x^5} + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac 1 {7 x^7} + \cdots}\right)       | c =  }} {{end-eqn}} which converges for $\left\lvert{x}\right\rvert \ge 1$.
Fuzzy Intersection is Commutative
Fuzzy intersection is commutative.
Integral Domain is Reduced Ring
Let $\left({D, +, \circ}\right)$ be an integral domain. Then $D$ is reduced.
Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 2/Proof 2
:$\vdash \left({p \land \left({q \lor r}\right)}\right) \iff \left({\left({p \land q}\right) \lor \left({p \land r}\right)}\right)$
Ring of Polynomial Functions is Commutative Ring with Unity
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $R \sqbrk {\set {X_j: j \in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\set {X_j: j \in J}$. Let $R^J$ be the free module on $J$. Let $A$ be the set of all polynomial functions $R^J \to R$. Let $\struct {A, +, \circ}$ be the ring of polynomial functions on $R$. Then $\struct {A, +, \circ}$ is a commutative ring with unity.
Moment Generating Function of Continuous Uniform Distribution
Let $X \sim \operatorname U \left[{a \,.\,.\, b}\right]$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then the moment generating function of $X$, $M_X$ is given by:  :$\displaystyle M_X \left({t}\right) = \begin{cases} \dfrac{ e^{t b} - e^{t a} } {t \left({b - a}\right)} & t \ne 0 \\ 1 & t = 0 \end{cases}$
Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign/Corollary
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = E_{2 n}       | r = \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {\paren {2 j + 1}^{2 n + 1} }       | c =  }} {{eqn | r = \paren {-1}^{n + 1} \dfrac {2^{2 n + 2} \paren {2 n}!} {\pi^{2 n + 1} } \paren {\frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots}       | c =  }} {{end-eqn}}
Natural Number has Same Prime Factors as Integer Power
Let $x$ be a natural number such that $x > 1$. Let $n \ge 1$ be a (strictly) positive integer. The $n$th power of $x$ has the same prime factors as $x$.
Equivalence of Definitions of Well-Ordering
{{TFAE|def = Well-Ordering}} Let $\left({S, \preceq}\right)$ be a ordered set.
Contour Integral of Gamma Function
Let $\Gamma$ denote the gamma function. Let $y$ be a positive number. Then for any positive number $c$: :$\displaystyle \frac 1 {2 \pi i} \int_{c - i \infty}^{c + i \infty} \Gamma \left({t}\right) y^{-t} \rd t = e^{-y}$
Sign of Composition of Permutations
Let $n \in \N$ be a natural number. Let $N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $S_n$ denote the set of permutations on $N_n$. Let $\map \sgn \pi$ denote the sign of $\pi$ of a permutation $\pi$ of $N_n$. Let $\pi_1, \pi_2 \in S_n$. Then: :$\map \sgn {\pi_1} \, \map \sgn {\pi_2} = \map \sgn {\pi_1 \circ \pi_2}$ where $\pi_1 \circ \pi_2$ denotes the composite of $\pi_1$ and $\pi_2$.
Relational Closure from Transitive Closure
Let $A$ be a set or class. Let $\RR$ be a relation on $A$. Let $\RR^+$ be the transitive closure of $\RR$. Let $B \subseteq A$. Let $B' = B \cup \map {\paren {\RR^+}^{-1} } B$. Let $C$ be an $\RR$-transitive subset or subclass of $A$ such that $B \subseteq C$. Then: :$B'$ is $\RR$-transitive :$B' \subseteq C$ :If $B$ is a set and $\RR$ is set-like then $B'$ is a set. That is, $B'$ is the relational closure of $B$ under $\RR$.
Product of r Choose m with m Choose k/Complex Numbers
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers: :$\dbinom z t \dbinom t w = \dbinom z w \dbinom {z - w} {t - w}$ where $\dbinom z w$ is a binomial coefficient.
Approximation to 2n Choose n
:$\displaystyle \lim_{n \mathop \to \infty} \dbinom {2 n} n = \dfrac {4^n} {\sqrt {n \pi} }$
Upper Bound for Harmonic Number
:$H_{2^m} \le 1 + m$ where $H_{2^m}$ denotes the $2^m$th harmonic number.
Smallest Strictly Positive Rational Number does not Exist
There exists no smallest element of the set of strictly positive rational numbers.
Generating Function for Powers of Two
Let $\sequence {a_n}$ be the sequence defined as: :$\forall n \in \N: a_n = 2^n$ That is: :$\sequence {a_n} = 1, 2, 4, 8, \ldots$ Then the generating function for $\sequence {a_n}$ is given as: :$\displaystyle \map G z = \frac 1 {1 - 2 z}$ for $\size z < \dfrac 1 2$
Exchange of Order of Summation with Dependency on Both Indices/Example
Let $n \in \Z$ be an integer. Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be the propositional function on the set of integers defining: :$\forall i \in \Z: \map R 1 := \paren {n = k i \text { for some } k \in \Z}$ Let $S: \Z \times \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the Cartesian product of the set of integers with itself defining: :$\forall i, j \in \Z: \map S {i, j} := \paren {1 \le j < i}$ Consider the summation: :$\displaystyle \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}$ Then: :$\displaystyle \sum_{\map R i} \sum_{\map S {i, j} } a_{i j} = \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}$ where: :$\map {S'} j$ denotes the propositional function: ::$\forall j \in \Z: \map {S'} j := \paren {1 < j \le n}$ :$\map {R'} {i, j}$ denotes the propositional function: ::$\forall i, j \in \Z: \map {R'} {i, j} := \paren {n = k i \text { for some } k \in \Z \text { and } i > j}$
Probability Generating Function defines Probability Distribution
Let $X$ and $Y$ be discrete random variables  whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$. Let the probability generating functions of $X$ and $Y$ be $\map {\Pi_X} s$ and $\map {\Pi_Y} s$ respectively. Then: :$\forall s \in \closedint {-1} 1: \map {\Pi_X} s = \map {\Pi_Y} s$ : :$\forall k \in \N: \Pr \left({X = k}\right) = \map \Pr {Y = k}$ That is, discrete random variables which are integer-valued have the same PGFs  they have the same PMF.
Expectation of Square of Discrete Random Variable
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \paren s$. Then the square of the expectation of $X$ is given by the expression: :$\expect {X^2} = \Pi''_X \paren 1 + \Pi'_X \paren 1$ where $\Pi''_X \paren 1$ and $\Pi'_X \paren 1$ denote the second and first derivative respectively of the PGF $\Pi_X \paren s$ evaluated at $1$.
Modulo Operation/Examples/100 mod 7
:$100 \bmod 7 = 2$
Expectation and Variance of Poisson Distribution equal its Parameter
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the expectation of $X$ equals the variance of $X$, that is, $\lambda$ itself.
Modulo Operation as Integer Difference by Quotient
Let $x, y, z \in \R$ be real numbers. Let $y > 0$. Let $0 \le z < y$. Let: :$\dfrac {x - z} y = k$ for some integer $k$. Then: :$z = x \bmod y$ where $\bmod$ denotes the modulo operation.
Dirichlet Series Convergence Lemma/General
Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n \left({s}\right)}$ be a general Dirichlet series. Let $f \left({s}\right)$ converge at $s_0 = \sigma_0 + i t_0$. Then $f \left({s}\right)$ converge for all $s = \sigma + i t$ where $\sigma > \sigma_0$.
Second 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 second derivative of the PGF of $X$ {{WRT|Differentiation}} $s$ is: :$\dfrac {\mathrm d^2} {\mathrm d s^2} \Pi_X \left({s}\right) = \dfrac {n \left({n + 1}\right) p^2} {q^2} \left({\dfrac q {1 - p s} }\right)^{n + 2}$ where $q = 1 - p$.
Stirling's Formula/Refinement
A refinement of Stirling's Formula is: :$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} }$
Legendre's Theorem/Corollary
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $B$ be the binary representation of $n$. Let $r$ be the number of unit digits in $B$. Let $n!$ denote the factorial of $n$. Then $2^{n - r}$ is a divisor of $n!$, but $2^{n - r + 1}$ is not.
Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind
Let $z \in \C$ be a complex number whose real part is positive. Then: :$z^{\underline r} = \displaystyle \sum_{k \mathop = 0}^m \left[{r \atop r - k}\right] \left({-1}\right)^k z^{r - k} + \mathcal O \left({z^{r - m - 1} }\right)$ where: :$\displaystyle \left[{r \atop r - k}\right]$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane :$z^{\underline r}$ denotes $z$ to the $r$ falling :$\mathcal O \left({z^{r - m - 1} }\right)$ denotes big-$\mathcal O$ notation.
Galois Field of Order q Exists iff q is Prime Power
Let $q \ge 0$ be a positive integer. Then there exists a Galois field of order $q$  $q$ is a prime power.
Product of r Choose k with r Minus Half Choose k/Formulation 2
Let $k \in \Z$, $r \in \R$. :$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} {2 k} \dbinom {2 k} k} {4^k}$ where $\dbinom r k$ denotes a binomial coefficient.
Existence of Minimal Uncountable Well-Ordered Set
There exists a minimal uncountable well-ordered set. That is, there exists an uncountable well-ordered set $\Omega$ with the property that every initial segment in $\Omega$ is countable.