Uncountable Product of First-Countable Spaces is not always First-Countable

Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a first-countable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a first-countable space.


Uncountable Product of Separable Spaces is not always Separable

Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a separable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a separable space.


Product of Metacompact Spaces is not always Metacompact

Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a metacompact space. Then it is not necessarily the case that $\struct {S, \tau}$ is also metacompact space.


Finite Irreducible Space is Path-Connected

Let $T = \left({S, \tau}\right)$ be a finite irreducible topological space. Then $T$ is path-connected.


Equivalence of Definitions of Convergent of Continued Fraction

Let $F$ be a field, such as the field of real numbers. Let $n \in \N \cup \set \infty$ be an extended natural number. Let $C = \sqbrk {a_0, a_1, a_2, \dotsc}$ be a continued fraction in $F$ of length $n$. Let $k \le n$ be a natural number. {{TFAE|def = Convergent of Continued Fraction}}


Locally Connected Space is not necessarily Locally Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also an locally patj-connected space.


Binary Product in Preadditive Category is Biproduct

Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \times a_2, p_1, p_2)$ be their binary product, assuming it exists. Let $i_1 : a_1 \to a_1 \times a_2$ be the unique morphism with: :$p_1 \circ i_1 = 1 : a_1 \to a_1$ :$p_2 \circ i_1 = 0 : a_1 \to a_2$ Let $i_2 : a_1 \to a_1 \times a_2$ be the unique morphism with: :$p_1 \circ i_2 = 0 : a_2 \to a_1$ :$p_2 \circ i_2 = 1 : a_2 \to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \times a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.


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 $S$ {{iff}} $\set {\relcomp S X: X \in \mathcal C}$ 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 $T$ {{iff}} 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 {{iff}} $T$ 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 {{iff}} $T$ 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 $A$. {{TFAE}} #$C$ 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: :$\forall s \in \closedint {-1} 1: \map {\Pi_X} s = \map {\Pi_Y} s$ {{iff}}: :$\forall k \in \N: \Pr \left({X = k}\right) = \map \Pr {Y = k}$ That is, discrete random variables which are integer-valued have the same PGFs {{iff}} 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 $q$ {{iff}} $q$ 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.


Inverse of Stirling's Triangle expressed as Matrix

Consider Stirling's triangle of the first kind (signed) expressed as a (square) matrix $\mathbf A$, with the top left element holding $s \left({0, 0}\right)$. :$\begin{pmatrix} 1 &     0 &       0 &      0 &      0 &     0 & \cdots \\ 0 &     1 &       0 &      0 &      0 &     0 & \cdots \\ 0 &    -1 &       1 &      0 &      0 &     0 & \cdots \\ 0 &     2 &      -3 &      1 &      0 &     0 & \cdots \\ 0 &    -6 &      11 &     -6 &      1 &     0 & \cdots \\ 0 &    24 &     -50 &     35 &    -10 &     1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &  \ddots \\ \end{pmatrix}$ Thenconsider Stirling's triangle of the second kind expressed as a (square) matrix $\mathbf B$, with the top left element holding $\displaystyle \left \{ {0 \atop 0}\right\}$. :$\begin{pmatrix} 1 & 0 &   0 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   0 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   1 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   3 &    1 &    0 &    0 & \cdots \\ 0 & 1 &   7 &    6 &    1 &    0 & \cdots \\ 0 & 1 &  15 &   25 &   10 &    1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$ Then: : $\mathbf A = \mathbf B^{-1}$ that is: : $\mathbf B = \mathbf A^{-1}$


Reflection Rule for Gaussian Binomial Coefficients

Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k_q = q^{k \paren {n - k} } \dbinom n k_{q^{-1} }$ where $\dbinom n k_q$ is a Gaussian binomial coefficient.


Rooted Tree Corresponds to Arborescence

Let $T = \struct {V, E}$ be a rooted tree with root $r$. Then there is a unique orientation of $T$ which is an $r$-arborescence.


Equivalence of Definitions of Arborescence

Let $G = \struct {V, A}$ be a directed graph. Let $r \in V$. {{TFAE|def = Arborescence}}


Inductive Construction of Sigma-Algebra Generated by Collection of Subsets

Let $\EE$ be a set of sets which are subsets of some set $X$. Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$. Then $\map \sigma \EE$  can be constructed inductively. The construction is as follows: Let $\Omega$ denote the minimal uncountable well-ordered set. Let $\alpha$ be an arbitrary initial segment in $\Omega$. Considering separately the cases whether or not $\alpha$ has an immediate predecessor $\beta$, we define: :$\EE_1 = \EE$ :$\EE_\alpha = \begin{cases} \set {\SS \in \powerset {\EE_\beta}: \SS \text { is countable or } \SS^\complement \text{ is countable} } & \alpha \text{ has an immediate predecessor } \beta \\ \ds \bigcup_{\beta \mathop \prec \alpha} \EE_\beta & \text { otherwise} \end{cases}$ :$\EE_\Omega = \ds \bigcup_{\alpha \mathop \in \Omega} \EE_\alpha$ Then $\map \sigma \EE = \EE_\Omega$.


Equivalence of Definitions of Set Partition

Let $S$ be a set {{TFAE|def = Set Partition}}


Summation to n of Reciprocal of k by k-1 of Harmonic Number

:$\displaystyle \sum_{1 \mathop < k \mathop \le n} \dfrac 1 {k \paren {k - 1} } H_k = 2 - \dfrac {H_n} n - \dfrac 1 n$ where $H_n$ denotes the $n$th harmonic number.


Riemann Zeta Function of 1000

To at least $100$ decimal places: :$\zeta \left({1000}\right) \approx 1$ where $\zeta$ denotes the Riemann zeta function.


Logarithmic Integral as Non-Convergent Series

The logarithmic integral can be defined in terms of a non-convergent series. That is: :$\displaystyle \operatorname {li} \left({z}\right) = \sum_{i \mathop = 0}^{+\infty} \frac {i! \, z} {\ln^{i + 1} z}  = \frac z {\ln z} \left({\sum_{i \mathop = 0}^{+\infty} \frac {i!} {\ln^i z} }\right)$


Bridge divides Graph into Two Components

Let $G$ be a connected graph. Let $e$ be a bridge of $G$. Then the edge deletion $G - e$ contains exactly $2$ components.


Connected Graph with only Even Vertices has no Bridge

Let $G$ be a connected graph whose vertices are all even. Then no edge of $G$ is a bridge.


Cut-Vertex divides Graph into Two or More Components

Let $G$ be a graph. Let $v$ be a cut-vertex of $G$. Then the vertex deletion $G - v$ contains $2$ or more components.


Median to Hypotenuse of Right Triangle equals Half Hypotenuse

Let $\triangle ABC$ be a right triangle such that $BC$ is the hypotenuse. Let $AD$ be the median to $BC$. Then $AD$ is half of $BC$.


Summation to n of Square of kth Harmonic Number

:$\displaystyle \sum_{k \mathop = 1}^n {H_k}^2 = \paren {n + 1} {H_n}^2 - \paren {2 n + 1} H_n + 2 n$ where $H_k$ denotes the $k$th harmonic number.


1+1 = 2

Define $0$ as the only element in the set $P \setminus s \left({P}\right)$, where: :$P$ is the Peano Structure :$s \left({P}\right)$ is the image of the mapping $s$ defined in Peano structure : $\setminus$ denotes the set difference. The theorem to be proven is: :$1 + 1 = 2$ where: :$1 := s \left({0}\right)$ :$2 := s \left({1}\right) = s \left({s \left({0}\right)}\right)$ :$+$ denotes addition :$=$ denotes equality :$s \left({n}\right)$ denotes the successor function as defined by Peano


Highest Power of 2 Dividing Numerator of Sum of Odd Reciprocals

Let: : $S = \dfrac p q = \displaystyle \sum_{k \mathop = 1}^n \dfrac 1 {2 k - 1}$ where $\dfrac p q$ is the canonical form of $S$. Let $n = 2^k m$ where $m$ is odd. Then the largest power of $2$ that divides $p$ is $2^{2 k}$.


Two Paths between Vertices in Cycle Graph

Let $G$ be a simple graph. Let $u, v$ be vertices in $G$ such that $u \ne v$. Then: :for any two vertices $u, v$ in $G$ such that $u \ne v$ there exists exactly two paths between $u$ and $v$ {{iff}}: :$G$ is a cycle graph.


Value of Radian in Degrees

The value of a radian in degrees is given by: :$1 \radians = \dfrac {180 \degrees} {\pi} \approx 57.29577 \ 95130 \ 8232 \ldots \degrees$  {{OEIS|A072097}}


Cassini's Identity/Negative Indices

Let $n \in \Z_{<0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then Cassini's Identity: :$F_{n + 1} F_{n - 1} - F_n^2 = \left({-1}\right)^n$ continues to hold.


Sum over k of n Choose k by Fibonacci Number with index m+k

:$\displaystyle \sum_{k \mathop \ge 0} \binom n k F_{m + k} = F_{m + 2 n}$ where: : $\dbinom n k$ denotes a binomial coefficient : $F_n$ denotes the $n$th Fibonacci number.


Area of Regular Polygon

Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the area of $P$ is given by: :$\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$ where $\cot$ denotes cotangent.


Fibonacci Number whose Index is Plus or Minus Prime p is Multiple of p

Let $p$ be a prime number distinct from $5$. Let $F_n$ denote the $n$th Fibonacci number. Then either $F_{p - 1}$ or $F_{p + 1}$ (but not both) is a multiple of $p$.


Perimeter of Regular Polygon by Circumradius

Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\mathcal P$ of $P$ is given by: :$\mathcal P = 2 n r \sin \dfrac \pi n$


Area of Regular Polygon by Inradius

Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the area $\AA$ of $P$ is given by: :$\AA = n r^2 \tan \dfrac \pi n$


Zeckendorf Representation of Integer shifted Right

Let $f \left({x}\right)$ be the real function defined as: :$\forall x \in \R: f \left({x}\right) = \left\lfloor{x + \phi^{-1} }\right\rfloor$ where: :$\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function :$\phi$ denotes the golden mean. Let $n \in \Z_{\ge 0}$ be a positive integer. Let $n$ be expressed in Zeckendorf representation: :$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$ with the appropriate restrictions on $k_1, k_2, \ldots, k_r$. Then: :$F_{k_1 - 1} + F_{k_2 - 1} + \cdots + F_{k_r - 1} = f \left({\phi^{-1} n}\right)$


Linear Combination of Generating Functions

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. and $H \left({z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$. Then $\alpha G \left({z}\right) + \beta H \left({z}\right)$ is the generating function for the sequence $\left\langle{\alpha a_n + \beta b_n}\right\rangle$.


Semigroup is Group Iff Latin Square Property Holds

Let $\left({S, \circ}\right)$ be a semigroup. Then $\left({S, \circ}\right)$ is a group {{iff}} for all $a, b \in S$ the Latin square property holds in $S$: :$a \circ x = b$ :$y \circ a = b$ for $x$ and $y$ each unique in $S$.


Generating Function for Even Terms of Sequence

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Consider the subsequence $\left\langle{b_n}\right\rangle := \left({a_0, a_2, a_4, \ldots}\right)$ Then the generating function for $\left\langle{b_n}\right\rangle$ is: :$\dfrac 1 2 \left({G \left({z}\right) + G \left({-z}\right)}\right)$


Ultraproduct is Well-Defined

'''Definition:Ultraproduct is well-defined.''' More specificly, following the definitions on Definition:Ultraproduct, we are going to prove that: :(1) $f^\mathcal M$ is well-defined :(2) $R^\mathcal M$ is well-defined


Equation of Circular Arc in Complex Plane

Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane. Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane. Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$. Then the equation: :$\arg \dfrac {z - b} {z - a} = \lambda$ represents the arc of a circle with $AB$ as a chord subtending an angle $\lambda$ at $Z$ on the circumference.


Exponential of Complex Number plus 2 pi i

:$\map \exp {z + 2 \pi i} = \map \exp z$


Sextuple Angle Formula for Sine

:$\dfrac {\map \sin {6 \theta} } {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$


Real Complex Roots of Unity for Odd Index

Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is odd. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \in U_n$ such that $x \in \R$ is: :$x = 1$ That is, $1$ is the only complex $n$th root of unity which is a real number.


Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle

Let $\epsilon_1, \epsilon_2, \epsilon_3$ be complex numbers embedded in the complex plane such that: :$\epsilon_1, \epsilon_2, \epsilon_3$ all have modulus $1$ :$\epsilon_1 + \epsilon_2 + \epsilon_3 = 0$ Then: :$\paren {\dfrac {\epsilon_2} {\epsilon_1} }^3 = \paren {\dfrac {\epsilon_3} {\epsilon_2} }^2 = \paren {\dfrac {\epsilon_1} {\epsilon_3} }^2 = 1$


Factorisation of x^(2n)-1 in Real Domain

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n} - 1 = \paren {z - 1} \paren {z + 1} \displaystyle \prod_{k \mathop = 1}^n \paren {z^2 - 2 \cos \dfrac {k \pi} n + 1}$


4 Sine Pi over 10 by Cosine Pi over 5

:$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$


Secant of Complement equals Cosecant

:$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$ where $\sec$ and $\csc$ are secant and cosecant respectively. That is, the cosecant of an angle is the secant of its complement. This relation is defined wherever $\sin \theta \ne 0$.


Cosecant of Complement equals Secant

:$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$ where $\csc$ and $\sec$ are cosecant and secant respectively. That is, the secant of an angle is the cosecant of its complement. This relation is defined wherever $\cos \theta \ne 0$.


Secant of Supplementary Angle

:$\map \sec {\pi - \theta} = -\sec \theta$ where $\sec$ denotes secant. That is, the secant of an angle is the negative of its supplement.


Absolutely Convergent Series is Convergent/Complex Numbers

Let $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ be an absolutely convergent series in $\C$. Then $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ is convergent.


Existence of Radius of Convergence of Complex Power Series/Absolute Convergence

Let $B_R \paren \xi$ denote the open $R$-ball of $\xi$. Let $z \in B_R \paren \xi$. Then $S \paren z$ converges absolutely. If $R = +\infty$, we define $B_R \paren \xi = \C$.


Imaginary Part of Complex Exponential Function

Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\exp z$ denote the complex exponential function. Then: :$\Im \paren {\exp z} = e^x \sin y$ where: :$\Im z$ denotes the imaginary part of a complex number $z$ :$e^x$ denotes the real exponential function of $x$ :$\sin y$ denotes the real sine function of $y$.


Modulus of Sine of Complex Number

Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\sin z$ denote the complex sine function. Then: :$\cmod {\sin z} = \sqrt {\sin^2 x + \sinh^2 y}$ where: :$\cmod z$ denotes the modulus of a complex number $z$ :$\sin x$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function.


Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0

Consider the circle $C$ embedded in the complex plane defined by the equation: :$x^2 + y^2 - 2 x - 2 y - 2 = 0$ Let $z = x + i y \in \C$ be a point lying on $C$. Then: :$e^{-1} \le \cmod {e^z} \le e^3$


Secant of Angle plus Three Right Angles

:$\map \sec {x + \dfrac {3 \pi} 2} = \csc x$


Cosecant of Angle plus Three Right Angles

:$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$


Secant of Three Right Angles less Angle

:$\map \sec {\dfrac {3 \pi} 2 - \theta} = -\csc \theta$ where $\sec$ and $\csc$ are secant and cosecant respectively.


Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary

:$B_n = \displaystyle \sum_{k \mathop = 1}^n {n \brace k}$


Structure Induced by Commutative Ring Operations is Commutative Ring

Let $\struct {R, +, \circ}$ be a commutative ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$. Then $\struct {R^S, +', \circ'}$ is a commutative ring.


Equivalence Relation on Power Set induced by Intersection with Subset/Cardinality of Set of Equivalence Classes

Let $A$ be finite with $\card A = n$, where $\card {\, \cdot \,}$ denotes cardinality. The cardinality of the set of $\alpha$-equivalence classes is given by: :$\card {\set {\eqclass X \alpha: X \in S} } = 2^n$


Cosine to Power of Even Integer

{{begin-eqn}} {{eqn | l = \cos^{2 n} \theta       | r = \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \paren {\cos 2 n \theta + \binom {2 n} 1 \map \cos {2 n - 2} \theta + \cdots + \binom {2 n} {n - 1} \cos 2 \theta}       | c =  }} {{eqn | r = \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \sum_{k \mathop = 0}^{n - 1} \binom {2 n} k \map \cos {2 n - 2 k} \theta       | c =  }} {{end-eqn}}


Law of Tangents

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: :$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$


Functionally Complete Singleton Sets

The only binary logical connectives that form singleton sets which are functionally complete are NAND and NOR.