hcju/proofwikips huggingface dataset:
Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets
Let $T = \struct {S, \tau}$ be a topological space. Let the components of $T$ be open sets. Then: :$S$ is a union of open connected sets of $T$.
Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open
Let $T = \struct {S, \tau}$ be a topological space. Let $S$ be a union of open connected sets of $T$. Then: :The components of $T$ are open sets.
Components are Open iff Union of Open Connected Sets/Lemma 1
::$U \cap C \neq \O$  $U \neq \O$ and $U \subseteq C$
Integral to Infinity of Square of Sine p x over x Squared
:$\displaystyle \int_0^\infty \paren {\frac {\sin p x} x}^2 \rd x = \frac {\pi \size p} 2$
Integral to Infinity of One minus Cosine p x over x Squared
:$\displaystyle \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x = \frac {\pi \size p} 2$
1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways
The number $1$ can be expressed as the sum of $4$ distinct unit fractions in $6$ different ways: {{begin-eqn}} {{eqn | l = 1       | r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 + \frac 1 {24} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 9 + \frac 1 {18} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 {10} + \frac 1 {15} }} {{eqn | r = \frac 1 2 + \frac 1 4 + \frac 1 5 + \frac 1 {20} }} {{eqn | r = \frac 1 2 + \frac 1 4 + \frac 1 6 + \frac 1 {12} }} {{end-eqn}}
Empty Set is Subset of Power Set
The empty set is a subset of all power sets: :$\forall S: \O \subseteq \powerset S$
Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2
Let $x$ have a neighborhood basis consisting of connected sets.
Equivalence of Definitions of Weakly Locally Connected at Point/Definition 2 implies Definition 1
Let every open neighborhood $U$ of $x$ contain an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$.
Cycle of Subsets implies Set Equality
Let $A_1, A_2, \dotsc, A_n$ be sets. Let: :$\forall k \in \set {2, 3, \dotsc, n}: A_{k - 1} \subseteq A_k$ and: :$A_n \subseteq A_1$ Then: :$\forall j, k \in \set {1, 2, \dotsc, n}: A_j = A_k$
Correspondence between Set and Ordinate of Cartesian Product is Mapping
Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the relation on $S \times {S \times T}$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Then $j_t$ is a mapping.
Mapping from Set to Ordinate of Cartesian Product is Injection
Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the mapping from $S$ to $S \times T$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Then $j_t$ is an injection.
Primitive of Sine Integral Function
:$\displaystyle \int \map \Si x \rd x = x \map \Si x + \cos x + C$
Preimage of Subset of Cartesian Product under Injection from Factor
Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the injection from $S$ to $S \times T$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Let $W \subseteq S \times T$. Let $V = {j_t}^{-1} \sqbrk W$ denote the preimage of $W$ under $j_t$. Then: :$V = \set {s: \tuple {s, t} \in W}$
Derivative of Error Function
:$\displaystyle \frac \d {\d x} \paren {\map \erf x} = \frac 2 {\sqrt \pi} e^{-x^2}$
Limit to Infinity of Error Function
:$\displaystyle \lim_{x \mathop \to \infty} \map \erf x = 1$
Dirichlet Beta Function in terms of Hurwitz Zeta Function
:$\displaystyle \map \beta s = \frac 1 {4^s} \paren {\map \zeta {s, \frac 1 4} - \map \zeta {s, \frac 3 4} }$
Inverse of Right-Total Relation is Left-Total
:$\RR$ is right-total  $\RR^{-1}$ is left-total.
Inverse of Left-Total Relation is Right-Total
:$\mathcal R$ is left-total  $\mathcal R^{-1}$ is right-total.
Inverse of Mapping is Right-Total Relation
Let $f$ be a mapping. Then its inverse $f^{-1}$ is a right-total relation.
Inverse of One-to-One Relation is One-to-One
The inverse of a one-to-one relation is a one-to-one relation.
Inverse of Injection is One-to-One Relation
Let $f$ be an injective mapping. Then its inverse $f^{-1}$ is a one-to-one relation.
Inverse of Surjection is Relation both Left-Total and Right-Total
Let $f$ be an surjective mapping. Then its inverse $f^{-1}$ is a relation which is both left-total and right-total.
Inverse is Mapping implies Mapping is Injection
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then $f$ is an injection.
Inverse is Mapping implies Mapping is Surjection
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then $f$ is a surjection.
Derivative Function is not Invertible
Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$. Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$. Let $d: B \to A$ denote the mapping defined as: :$\forall \map f x \in B: \map d f = \map {D_x} f$ where $D_x$ denotes the derivative of $f$ {{WRT|Differentiation}} $x$. Then $d$ is not an invertible mapping.
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4
Let $\sequence {x_k}$ satisfy: :$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$ where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2
Let $\sequence {x_k}$ satisfy: :for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.
Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x
:$\displaystyle \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
Map from Set of Continuous Functions on Interval to Set of their Integrals
Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$. Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$. Let $C \subseteq B$ denote the subset of $B$ which consists of all elements of $B$ such that $\map f a = 0$. For each $f \in A$, let $h$ denote the mapping defined as: :$\map {\paren {\map h f} } x = \displaystyle \int_a^x \map f t \rd t$ Then: :$h: A \to C$
Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x
:$\displaystyle \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$
Möbius Transformation is Bijection
Let $a, b, c, d \in \C$ be complex numbers. Let $f: \overline \C \to \overline \C$ be the Möbius transformation: :$\map f z = \begin {cases} \dfrac {a z + b} {c z + d} & : z \ne -\dfrac d c \\ \infty & : z = -\dfrac d c \\ \dfrac a c & : z = \infty \\ \infty & : z = \infty \text { and } c = 0 \end {cases}$ Then: :$f: \overline \C \to \overline \C$ is a bijection : :$a c - b d \ne 0$
Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x
:$\displaystyle \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$
Definite Integral from 0 to 1 of x to the x
{{begin-eqn}} {{eqn	| l = \int_0^1 x^x \rd x 	| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n} }} {{eqn	| r = -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n} }} {{eqn	| r = 0.78343 \ 05107 \ 12\ldots }} {{end-eqn}}
Definite Integral from 0 to 1 of x to the minus x
{{begin-eqn}} {{eqn	| l = \int_0^1 x^{-x} \rd x 	| r = \sum_{n \mathop = 1}^\infty n^{-n} }} {{eqn	| r = 1.29128 \ 5997 \ldots }} {{end-eqn}}
Definite Integral to Infinity of Exponential of -a x by Sine of b x
:$\displaystyle \int_0^\infty e^{-a x} \sin b x \rd x = \frac b {a^2 + b^2}$
Sum of Exponential of i k x
:$\displaystyle \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$
Extension of Extension of Mapping is Extension
Let $A, B, C, S$ be sets such that $A \subseteq B \subseteq C$. Let $f: A \to S$, $g: B \to S$ and $h: C \to S$ be mappings such that: :$g$ is an extension of $f$ to $B$ :$h$ is an extension of $g$ to $C$. Then $h$ is an extension of $f$ to $C$.
Indexed Cartesian Space is Set of all Mappings
Let $I$ be an indexing set. Let $\displaystyle \prod_{i \mathop \in I} S$ denote the cartesian space of $S$ indexed by $I$. Then $\displaystyle \prod_{i \mathop \in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation: :$S^I := \displaystyle \prod_{i \mathop \in I} S$
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2
Let $M = \struct {A, d}$ be a metric space or a pseudometric space. Let $\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}
Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3
Let $M = \struct {A, d}$ be a metric space or a pseudometric space. Let $\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}
Definite Integral from 0 to 1 of Logarithm of x over One minus x
:$\displaystyle \int_0^1 \frac {\ln x} {1 - x} \rd x = -\frac {\pi^2} 6$
Definite Integral from 0 to 1 of Logarithm of One plus x over x
:$\displaystyle \int_0^1 \frac {\map \ln {1 + x} } x \rd x = \frac {\pi^2} {12}$
Definite Integral from 0 to 1 of Logarithm of One minus x over x
:$\displaystyle \int_0^1 \frac {\map \ln {1 - x} } x \rd x = -\frac {\pi^2} 6$
Definite Integral to Infinity of Exponential of -a x by Cosine of b x
:$\displaystyle \int_0^\infty e^{-a x} \cos b x \rd x = \frac a {a^2 + b^2}$
Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x
:$\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x = \ln \frac b a$
Definite Integral to Infinity of Exponential of -a x^2
:$\displaystyle \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$
Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x
:$\displaystyle \int_0^1 \frac {x^m - x^n} {\ln x} \rd x = \map \ln {\frac {m + 1} {n + 1} }$
Union is Commutative/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $\displaystyle I = \bigcup_{i \mathop \in I} S_i$ denote the union of  $\family {S_i}_{i \mathop \in I}$. Let $J \subseteq I$ be a subset of $I$. Then: :$\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \mathop \in J} S_j$ where $\relcomp I J$ denotes the complement of $J$ relative to $I$.
Intersection is Commutative/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $\displaystyle I = \bigcap_{i \mathop \in I} S_i$ denote the intersection of $\family {S_i}_{i \mathop \in I}$. Let $J \subseteq I$ be a subset of $I$. Then: :$\displaystyle \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j$ where $\relcomp I J$ denotes the complement of $J$ relative to $I$.
Definite Integral to Infinity of Power of x by Logarithm of x over One plus x
:$\displaystyle \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$
Definite Integral from 0 to Half Pi of Logarithm of Sine x
:$\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$
Definite Integral from 0 to Half Pi of Logarithm of Cosine x
:$\displaystyle \int_0^{\pi/2} \map \ln {\cos x} \rd x = -\frac \pi 2 \ln 2$
Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0$. Then: :$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$
Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One
:$\displaystyle \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x = \frac {\pi^2} 4$
Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x
:$\displaystyle \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$
Primitive of Sine x by Logarithm of Sine x
:$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$
Euler-Mascheroni Constant as Difference of Integrals involving Cosine
:$\displaystyle \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$
Definite Integral from 0 to 1 of Arcsine of x over x
:$\displaystyle \int_0^1 \frac {\arcsin x} x = \frac \pi 2 \ln 2$
Definite Integral from 0 to 1 of Arctangent of x over x
:$\displaystyle \int_0^1 \frac {\arctan x} x \rd x = G$
Definite Integral from 0 to Half Pi of x over Sine x
:$\displaystyle \int_0^{\pi/2} \frac x {\sin x} \rd x = 2 G$
Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x
:$\displaystyle \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$
Definite Integral from 0 to Half Pi of Sine x by Logarithm of Sine x
:$\displaystyle \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$
Definite Integral to Infinity of Exponential of -(a x^2 plus b x plus c)
:$\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }$
Set is Subset of Intersection of Supersets
Let $S$, $T_1$ and $T_2$ be sets. Let $S$ be a subset of both $T_1$ and $T_2$. Then: :$S \subseteq T_1 \cap T_2$ That is: :$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$
Definite Integral to Infinity of Power of x by Exponential of -a x^2
:$\displaystyle \int_0^\infty x^m e^{-a x^2} \rd x = \frac {\map \Gamma {\paren {m + 1}/2} } {2 a^{\paren {m + 1}/2} }$
Integral Representation of Dirichlet Eta Function in terms of Gamma Function
:$\displaystyle \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$
Definite Integral to Infinity of x over Exponential of x plus One
:$\displaystyle \int_0^\infty \frac x {e^x + 1} \rd x = \frac {\pi^2} {12}$
Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x
:$\displaystyle \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$
Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x
:$\displaystyle \int_0^1 \ln x \map \ln {1 + x} \rd x = 2 - 2 \ln 2 - \frac {\pi^2} {12}$
Power Series Expansion for Error Function
:$\displaystyle \map \erf x = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$
Set is Subset of Intersection of Supersets/General Result
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Let $X$ be a set such that: :$\forall i \in I: X \subseteq S_i$ Then: :$X \subseteq \bigcup_{i \mathop \in I} S_i$ where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.
Error Function is Odd
:$\map \erf {-x} = -\map \erf x$
Error Function of Zero
:$\map \erf 0 = 0$
P-adic Expansion is a Cauchy Sequence in P-adic Norm
Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals numbers $\Q$. Let $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion. Then the sequence of partial sums of the series: :$\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ is a Cauchy sequence in the valued field  $\struct{\Q, \norm{\,\cdot\,}_p}$.
P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Characterization of Exponential Integral Function
:$\displaystyle \map \Ei x = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u$
Membership Relation is Not Reflexive
Let $\Bbb S$ be a set of sets in the context of pure set theory Let $\RR$ denote the membership relation on $\Bbb S$: :$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$ $\RR$ is not in general a reflexive relation.
Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence{x_n}$ be a sequence in the rational numbers $\Q$. Then $\sequence {x_n}$ is a Cauchy sequence in $\struct{\Q, \norm{\,\cdot\,}_p}$  $\sequence {x_n}$ is a Cauchy sequence in $\struct{\Q_p, \norm{\,\cdot\,}_p}$
Definite Integral to Infinity of Sine x over Root x
:$\displaystyle \int_0^\infty \frac {\sin x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$
Digamma Function of One Half
:$\displaystyle \map \psi {\frac 1 2} = -\gamma - 2 \ln 2$
Definite Integral to Infinity of Exponential of -x^2 by Logarithm of x
:$\displaystyle \int_0^\infty e^{-x^2} \ln x \rd x = -\frac {\sqrt \pi} 4 \paren {\gamma + 2 \ln 2}$
Definite Integral to Infinity of Cosine x over Root x
:$\displaystyle \int_0^\infty \frac {\cos x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$
Definite Integral to Infinity of Cube of Sine x over x Cubed
:$\displaystyle \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$
Equivalence of Definitions of Composition of Mappings
{{TFAE|def = Composition of Mappings}} Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.
P-adic Numbers are Generated Ring Extension of P-adic Integers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Then: :$Q_p = \Z_p \sqbrk {1 / p}$ where $\Z_p \sqbrk {1 / p}$ denotes the ring extension generated by $1 / p$.
Limit to Infinity of Exponential Integral Function
:$\displaystyle \lim_{x \mathop \to \infty} \map \Ei x = 0$
Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x
:$\displaystyle \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$
Fourier Series/Identity Function over Symmetric Range
Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the identity function on the open real interval $\openint {-\lambda} \lambda$: :$\forall x \in \openint {-\lambda} \lambda: \map f x = x$ The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can be given as: :$\map f x \sim \dfrac {2 \lambda} \pi \displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda$
Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image
Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f {\restriction_X}$ be the restriction of $f$ to $X$. Then: :$f {\restriction_X} = f \cap \paren {X \times \Img f}$ where: :$\Img f$ denotes the image of $f$, defined as: ::$\Img f = \set {t \in T: \exists s \in S: t = \map f s}$ :$X \times \Img f$ denotes the cartesian product of $X$ with $\Img f$.
Primitive of Reciprocal of One plus Fourth Power of x
:$\displaystyle \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$
Sine Integral Function is Odd
:$\displaystyle \map \Si {-x} = -\map \Si x$
Sine Integral Function of Zero
:$\map \Si 0 = 0$
Arctangent of Root 3 over 3
:$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$
Cardinality of Set Union/2 Sets
Let $S_1$ and $S_2$ be finite sets. Then: :$\card {S_1 \cup S_2} = \card {S_1} + \card {S_2} - \card {S_1 \cap S_2}$
Cardinality of Set Union/3 Sets
Let $S_1$, $S_2$ and $S_3$ be finite sets. Then: {{begin-eqn}}      {{eqn | l = \card {S_1 \cup S_2 \cup S_3}       | r = \card {S_1} + \card {S_2} + \card {S_3}       | c = }} {{eqn | o =        | ro= -       | r = \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}       | c = }} {{eqn | o =        | ro= +       | r = \card {S_1 \cap S_2 \cap S_3}       | c = }} {{end-eqn}}
Definite Integral to Infinity of Exponential of -i x^2
:$\displaystyle \int_0^\infty \map \exp {-i x^2} \rd x = \frac 1 2 \sqrt {\frac \pi 2} \paren {1 - i}$
P-adic Number times Integer Power of p is P-adic Integer
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Then: :$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$
Definite Integral to Infinity of Sine of a x^2
:$\ds \int_0^\infty \map \sin {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$
Definite Integral to Infinity of Cosine of a x^2
:$\displaystyle \int_0^\infty \map \cos {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$
Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One
:$\displaystyle \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x = \frac 1 {2 a} - \frac \pi {2 b} \cot \frac {a \pi} b$
Union with Intersection equals Intersection with Union iff Subset
Let $A$, $B$ and $C$ be sets. Then: :$\paren {A \cap B} \cup C = A \cap \paren {B \cup C} \iff C \subseteq A$
(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A
Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$. Then: :$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$ where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.
Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3
Let $A$, $B$ and $C$ be sets. Then: :$\paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A} = \paren {A \cup B} \cap \paren {B \cup C} \cap \paren {C \cup A}$
Definite Integral to Infinity of Cosine of a x over Hyperbolic Cosine of b x
:$\displaystyle \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x = \frac \pi {2 b} \sech \frac {a \pi} {2 b}$
Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x plus One
:$\displaystyle \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x = \frac \pi {2 b} \csc \frac {a \pi} b - \frac 1 {2 a}$
Intersection Complement of Set with Itself is Complement
Let $A$ and $B$ be subsets of a universal set $\Bbb U$. Let $\uparrow$ denote the operation on $A$ and $B$ defined as: :$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$. Then: :$A \uparrow A = \relcomp {\Bbb U} A$
Set Union expressed as Intersection Complement
Let $A$ and $B$ be subsets of a universal set $\Bbb U$. Let $\uparrow$ denote the operation on $A$ and $B$ defined as: :$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$. Then: :$A \cup B = \paren {A \uparrow A} \uparrow \paren {B \uparrow B}$
Element in Set iff Singleton in Powerset
Let $S$ be a set. Then: :$x \in S \iff \set x \in \powerset S$ where $\powerset S$ denotes the power set of $S$.
Set Consisting of Empty Set is not Empty
Let $S$ be the set defined as: :$S = \set \O$ Then $S$ is not the empty set. That is: :$\O \ne \set \O$
Elements of Ordered Pair do not Commute
Let $\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects. Let $\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$. Then: :$\tuple {a, b} \ne \tuple {b, a}$