hcju/proofwikips huggingface dataset:
Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties
Let $S$ be a set which has at least $3$ elements. Then it is possible to set up a relation $\circledcirc$ on $S$ which has any combination of the $3$ properties: :Reflexivity :Symmetry :Transitivity but this is not possible for a set which has fewer than $3$ elements.
Half Angle Formulas/Hyperbolic Tangent/Corollary 1
:$\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$
Even Integer Plus 5 is Odd
Let $x \in \Z$ be an even integer. Then $x + 5$ is odd.
Minimal Smooth Surface Spanned by Contour
Let $\map z {x, y}: \R^2 \to \R$ be a real-valued function. Let $\Gamma$ be a closed contour in $3$-dimensional Euclidean space. Suppose this surface is smooth for every $x$ and $y$. Then it has to satisfy the following Euler's equation: :$r \paren {1 + q^2} - 2 s p q + t \paren {1 + p^2} = 0$ where: {{begin-eqn}} {{eqn | l = p       | r = z_x }} {{eqn | l = q       | r = z_y }} {{eqn | l = r       | r = z_{xx} }} {{eqn | l = s       | r = z_{xy} }} {{eqn | l = t       | r = z_{yy} }} {{end-eqn}} with subscript denoting respective partial derivatives. In other words, its mean curvature has to vanish.
Principle of Induction applied to Interval of Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $\left[{p \,.\,.\, q}\right]$ be a closed interval of $\left({S, \circ, \preceq}\right)$. Let $T \subseteq \left[{p \,.\,.\, q}\right]$ such that the minimal element of $\left[{p \,.\,.\, q}\right]$ is in $T$. Let: : $x \in T: x \prec q \implies x \circ 1 \in T$ Then: : $T = \left[{p \,.\,.\, q}\right]$
Lowest Common Multiple of Integers with Common Divisor
Let $b, d \in \Z_{>0}$ be (strictly) positive integers Then: :$\lcm \set {a b, a d} = a \lcm \set {b, d}$ where: :$a \in \Z_{>0}$ :$\lcm \set {b, d}$ denotes the lowest common multiple of $m$ and $n$.
GCD of Sum and Difference of Integers
:$\gcd \set {a + b, a - b} \ge \gcd \set {a, b}$
Inverse Hyperbolic Cotangent is Odd Function
:$\map {\coth^{-1} } {-x} = -\coth^{-1} x$
Inverse Cosine of Imaginary Number
:$\cos^{-1} x = \pm \, i \cosh^{-1} x$
Inverse Tangent of Imaginary Number
:$\tan^{-1} \left({i x}\right) = i \tanh^{-1} x$
LCM of 3 Integers in terms of GCDs of Pairs of those Integers
Let $a, b, c \in \Z_{>0}$ be strictly positive integers. Then: :$\lcm \set {a, b, c} = \dfrac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$ where: :$\gcd$ denotes greatest common divisor :$\lcm$ denotes lowest common multiple :$d_1 = \gcd \set {a, b}$ :$d_2 = \gcd \set {b, c}$ :$d_3 = \gcd \set {a, c}$
Integer Multiplication is Well-Defined
Integer multiplication is well-defined.
Inverse Hyperbolic Secant of Imaginary Number
:$\sech^{-1} x = \pm \, i \sec^{-1} x$
Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient
Let $F_n$ denote the $n$th Fibonacci number. Then: {{begin-eqn}} {{eqn | lo= \forall n \in \Z_{>0}:       | l = F_{2 n}       | r = \sum_{k \mathop = 1}^n \dbinom n k F_k       | c =  }} {{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n - 1} + \dbinom n n F_n       | c =  }} {{end-eqn}} where $\dbinom n k$ denotes a binomial coefficient.
Cardano's Formula/Real Coefficients
:$(1): \quad$ If $D > 0$, then one root is real and two are complex conjugates. :$(2): \quad$ If $D = 0$, then all roots are real, and at least two are equal. :$(3): \quad$ If $D < 0$, then all roots are real and unequal.
Cardano's Formula/Trigonometric Form
Let $a, b, c, d \in \R$. Let the discriminant $D < 0$, where $D := Q^3 + R^2$. Then the solutions of $P$ can be expressed as: :$x_1 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3} - \dfrac b {3 a}$ :$x_2 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3 + \dfrac {2 \pi} 3} - \dfrac b {3 a}$ :$x_3 = 2 \sqrt {-Q} \map \cos {\dfrac \theta 3 + \dfrac {4 \pi} 3} - \dfrac b {3 a}$ where: : $\cos \theta = \dfrac R {\sqrt{-Q^3} }$
Triangle Inequality/Complex Numbers/General Result
Let $z_1, z_2, \dotsc, z_n \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: :$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$
Substitution for Equivalent Subformula is Equivalent
Let $\mathbf B$ a WFF of propositional logic. Let $\mathbf A, \mathbf A'$ be equivalent WFFs. Let $\mathbf A$ be a subformula of $\mathbf B$. Let $\mathbf B' = \mathbf B \left({\mathbf A \,//\, \mathbf A'}\right)$ be the substitution of $\mathbf A'$ for $\mathbf A$ in $\mathbf B$. Then $\mathbf B$ and $\mathbf B'$ are equivalent.
Congruence Modulo Power of p as Linear Combination of Congruences Modulo p
Let $p$ be a prime number. Let $S = \set {a_1, a_2, \ldots, a_p}$ be a complete residue system modulo $p$. Then for all integers $n \in \Z$ and non-negative integer $s \in \Z_{\ge 0}$, there exists a congruence of the form: :$n \equiv \displaystyle \sum_{j \mathop = 0}^s b_j p^j \pmod {p^{s + 1} }$ where $b_j \in S$.
Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 62 Cards
Let $D$ be a deck of $62$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.
Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards
Let $D$ be a deck of $12$ cards. Let $D$ be given a sequence of modified perfect faro shuffles. Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in.
Exclusive Or with Tautology
:$p \oplus \top \dashv \vdash \neg p$
Difference of Two Even-Times Odd Powers
Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on. Let $n \in \Z_{> 0}$ be a (strictly) positive odd integer. Then: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \paren {a - b} \paren {a + b} \paren {\sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j} \paren {\sum_{j \mathop = 0}^{n - 1} \paren {-1}^j a^{n - j - 1} b^j}       | c =  }} {{eqn | r = \paren {a - b} \paren {a + b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} } \paren {a^{n - 1} - a^{n - 2} b + a^{n - 3} b^2 - \dotsb - a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}
Existence of Real Polynomial with no Real Root
There exist polynomials in real numbers $\R$ which have no roots in $\R$.
GCD from Congruence Modulo m
Let $a, b \in \Z, m \in \N$.   Let $a$ be congruent to $b$ modulo $m$. Then the GCD of $a$ and $m$ is equal to the GCD of $b$ and $m$. That is: :$a \equiv b \pmod m \implies \gcd \set {a, m} = \gcd \set {b, m}$
Equivalence of Definitions of Closed Set in Metric Space
{{TFAE|def = Closed Set (Metric Space)|view = Closed Set|context = Metric Space|contextview = Metric Spaces}} Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$.
Bolzano-Weierstrass Theorem/General Form
Every infinite bounded space in a real Euclidean space has at least one limit point.
Sum of Euler Phi Function over Divisors
Let $n \in \Z_{>0}$ be a strictly positive integer. Then $\displaystyle \sum_{d \mathop \divides n} \map \phi d = n$ where: :$\displaystyle \sum_{d \mathop \divides n}$ denotes the sum over all of the divisors of $n$ :$\map \phi d$ is the Euler $\phi$ function, the number of integers less than $d$ that are prime to $d$. That is, the total of all the totients of all divisors of a number equals that number.
Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1
:$\OO$ is a local ring.
Bessel's Correction
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :$\displaystyle \hat {\sigma^2} = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ is an unbiased estimator of $\sigma^2$.
Sum of Cosines of Fractions of Pi
Let $n \in \Z$ such that $n > 1$. Then: :$\displaystyle \sum_{k \mathop = 1}^{n - 1} \cos \frac {2 k \pi} n = -1$
Area of Quadrilateral in Determinant Form
Let $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$, $C = \tuple {x_3, y_3}$ and $D = \tuple {x_4, y_4}$ be points in the Cartesian plane. Let $A$, $B$, $C$ and $D$ form the vertices of a quadrilateral. The area $\mathcal A$ of $\Box ABCD$ is given by: :$\mathcal A = \dfrac 1 2 \paren {\size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } } + \size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_4 & y_4 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } } }$
Equation for Perpendicular Bisector of Two Points
Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two points in the cartesian plane. Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane. $L$ can be expressed by the equation: :$y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} 2}$
Finite Order Elements of Infinite Abelian Group form Normal Subgroup
Let $G$ be an infinite  abelian group. Let $H \subseteq G$ be the subset of $G$ defined as: :$H := \set {x \in G: x \text { is of finite order in } G}$ Then $H$ forms a normal subgroup of $G$.
Subgroup of Index 3 does not necessarily contain all Cubes of Group Elements
Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $3$. Then it is not necessarily the case that: :$\forall x \in G: x^3 \in H$
Complex Cross Product Distributes over Addition
Let $z_1, z_2, z_3 \in \C$ be complex numbers. Then: :$z_1 \times \paren {z_2 + z_3} = z_1 \times z_2 + z_1 \times z_3$ where $\times$ denotes cross product.
Finite Union of Open Sets in Complex Plane is Open
Let $S_1, S_2, \ldots, S_n$ be open sets of $\C$. Then $\displaystyle \bigcup_{k \mathop = 1}^n S_k$ is an open set of $\C$.
Limit Point of Set in Complex Plane not Element is Boundary Point
Let $S \subseteq \C$ be a subset of the complex plane. Let $z \in \C$ be a limit point of $S$ such that $z \notin S$. Then $z$ is a boundary point of $S$.
Order of Isomorphic Image of Group Element
Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$. Let $\phi: G \to H$ be a group isomorphism. Then: :$a \in G \implies \order {\map \phi a} = \order a$
Ring is Subring of Itself
Let $R$ be a ring. Then $R$ is a subring of itself.
Cyclic Group Elements whose Powers equal Identity
Let $G$ be a cyclic group whose identity is $e$ and whose order is $n$. Let $d \divides n$. Then there exist exactly $d$ elements $x \in G$ satisfying the equation $x^d = e$. These are the elements of the group $G_d$ generated by $g^{n / d}$:  :$G_d = \gen {g^{n / d} }$
Conditional iff Biconditional of Antecedent with Conjunction
:$p \implies q \dashv \vdash p \iff \left({p \land q}\right)$
Element in Integral Domain is Unit iff Principal Ideal is Whole Domain
:$x \in U_D \iff \ideal x = D$
Conjunction iff Biconditional of Biconditional with Disjunction
:$p \land q \dashv \vdash \left({p \iff q}\right) \iff \left({p \lor q}\right)$
Group Generated by Normal Intersection is Normal
Let $I$ be an indexing set, and $\left\{{N_i: i \in I}\right\}$ be a set of normal subgroups of the group $G$. Then $\left \langle {N_i: i \in I} \right \rangle$ is a normal subgroup of $G$.
Abelian Quotient Group
Let $G$ be a group. Let $H$ be a normal subgroup of $G$. Let $G / H$ denote the quotient group of $G$ by $H$. Then $G / H$ is abelian  $H$ contains every element of $G$ of the form $a b a^{-1} b^{-1}$ where $a, b \in G$.
Generator of Quotient Groups
Let $N \lhd G$ be a normal subgroup of $G$. Let: :$N \le A \le G$ :$N \le B \le G$ For a subgroup $H$ of $G$, let $\alpha$ be the bijection defined as: :$\map \alpha H = \set {h N: h \in H}$ Then: :$\map \alpha {\gen {A, B} } = \gen {\map \alpha A, \map \alpha B}$ where $\gen {A, B}$ denotes the subgroup of $G$ generated by $\set {A, B}$.
Inner Automorphism Group is Isomorphic to Quotient Group with Center
Let $G$ be a group. Let $\Inn G$ be the inner automorphism group of $G$. Let $\map Z G$ be the center of $G$. Let $G / \map Z G$ be the quotient group of $G$ by $\map Z G$. Then $G / \map Z G \cong \Inn G$.
Real Natural Logarithm is Restriction of Complex Natural Logarithm
Let $\ln: \C_{\ne 0} \to \C$ be the complex natural logarithm. Let $\ln': \R_{>0} \to \R$ be the real natural logarithm. Then: :$\ln' = \ln \restriction_{\R_{>0} \times \R}$ That is, the real natural logarithm is the restriction of the complex natural logarithm.
Inverse of Central Unit of Ring is in Center
Let $R$ be a ring. Let $\map Z R$ denote the center of $R$. Let $u \in R$ be a unit of $R$. Then: :$u \in \map Z R \implies u^{-1} \in \map Z R$
Commutativity of Group Direct Product
Let $\struct {G, \circ_g}$ and $\struct {H, \circ_h}$ be groups. Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_g}$ and $\struct {H, \circ_h}$, where the operation $\circ$ is defined as: :$\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_g g_2, h_1 \circ_h h_2}$ Let $\struct {H \times G, \star}$ be the group direct product of $\struct {H, \circ_h}$ and $\struct {G, \circ_g}$, where the operation $\star$ is defined as: :$\tuple {h_1, g_1} \star \tuple {h_2, g_2} = \tuple {h_1 \circ_h h_2, g_1 \circ_g g_2}$ The group direct product $\struct {G \times H, \circ}$ is isomorphic to the $\struct {H \times G, \star}$.
Associativity of Group Direct Product
The group direct product $G \times \paren {H \times K}$ is (group) isomorphic to $\paren {G \times H} \times K$.
Cyclic Group of Order 6
Let $C_n$ be the cyclic group of order $n$. Then: : $C_2 \times C_3 \cong C_6$ : $C_6$ is the internal group direct product of $C_2$ and $C_3$.
Direct Product of Central Subgroup with Inverse Isomorphism is Central Subgroup
Let $G$ and $H$ be groups. Let $\map Z G$ denote the center of $G$. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \cong W$ where $\cong$ denotes isomorphism. Let such a group isomorphism be $\theta: Z \to W$. Let $X$ be the set defined as: :$X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z}$ Then $X$ is a central subgroup of $G \times H$.
Non-Commutative Ring with Unity and 2 Ideals not necessarily Division Ring
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {R, +, \circ}$ specifically not be commutative. Let $\struct {R, +, \circ}$ be such that the only ideals of $\struct {R, +, \circ}$ are $\set {0_R}$ and $R$ itself. Then it is not necessarily the case that $\struct {R, +, \circ}$ is a division ring.
Equivalence of Definitions of Complex Inverse Hyperbolic Tangent
{{TFAE|def = Complex Inverse Hyperbolic Tangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$
Natural Numbers form Inductive Set
Let $\N$ denote the natural numbers as subset of the real numbers $\R$. Then $\N$ is an inductive set.
Additive Group of Reals is Normal Subgroup of Complex
Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\R, +}$ is a normal subgroup of $\struct {\C, +}$.
Generators of Additive Group of Integers
The only generators of the additive group of integers $\struct {\Z, +}$ are $1$ and $-1$.
Prime Power Mapping on Galois Field is Automorphism
Let $\GF$ be a Galois field whose zero is $0_\GF$ and whose characteristic is $p$. Let $\sigma: \GF \to \GF$ be defined as: :$\forall x \in \GF: \map \sigma x = x^p$ Then $\sigma$ is an automorphism of $\GF$.
Principal Ideal Domain cannot have Infinite Strictly Increasing Sequence of Ideals
Let $\struct {D, +, \circ}$ be a principal ideal domain. Then $D$ cannot have an infinite sequence of ideals $\sequence {j_n}_{n \mathop \in \N}$ such that: :$\forall n \in \N: J_n \subsetneq j_{n + 1}$
Field Norm of Complex Number is not Norm
Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is not a norm on $\C$.
Internal Angles of Square
The internal angles of a square are right angles.
Ideal of Ring of Polynomials over Field has Unique Monic Polynomial forming Principal Ideal
Let $F$ be a field. Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $J$ be a non-null ideal of $F \sqbrk X$. Then there exists exactly one monic polynomial $f \in F \sqbrk X$ such that: :$J = \ideal f$ where $\ideal f$ is the principal ideal generated by $f$ in $F \sqbrk X$.
Fixed Elements form 1-Cycles
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\Fix \pi$ be the set of elements fixed by $\pi$. For any $\pi \in S_n$, all the elements of $\Fix \pi$ form $1$-cycles.
Polynomial X^2 + 1 is Irreducible in Ring of Real Polynomials
Let $\R \sqbrk X$ be the ring of polynomials in $X$ over the real numbers $\R$. Then the polynomial $X^2 + 1$ is an irreducible element of $\R \sqbrk X$.
Equivalence of Definitions of Concave Real Function
Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Concave Real Function}}
Ring Subtraction equals Zero iff Elements are Equal
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ Then: :$\forall a, b \in R: a - b = 0_R \iff a = b$ where $a - b$ denotes ring subtraction.
Powers of Disjoint Permutations
Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho, \sigma$ be disjoint permutations. Then: : $\forall k \in \Z: \paren {\sigma \rho}^k = \sigma^k \rho^k$
Equivalence of Definitions of Strictly Concave Real Function
Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Concave Real Function}}
Equivalence of Definitions of Strictly Convex Real Function
Let $f$ be a real function which is defined on a real interval $I$. {{TFAE|def = Strictly Convex Real Function}}
Group Action on Subgroup of Symmetric Group
Let $S_n$ be the symmetric group of $n$ elements. Let $H$ be a subgroup of $S_n$. Let $X$ be any set with $n$ elements. Then $H$ acts on $X$ as a group of transformations on $X$.
Kernel of Group Action is Normal Subgroup
Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. Let $G_0$ denote the kernel of $\phi$. Then $G_0$ is a normal subgroup of $G$.
Equivalence of Definitions of Well-Ordered Integral Domain
{{TFAE|def = Well-Ordered Integral Domain}} Let $\struct {D, +, \times \le}$ be an ordered integral domain whose zero is $0_D$.
Second Principle of Finite Induction/Zero-Based
Let $S \subseteq \N$ be a subset of the natural numbers. Suppose that: :$(1): \quad 0 \in S$ :$(2): \quad \forall n \in \N: \paren {\forall k: 0 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$S = \N$
Integral of Exponent of Half Square over Reals
:$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} e^{- x^2 / 2} \rd x = \sqrt {2 \pi}$
Cayley's Representation Theorem
Let $S_n$ denote the symmetric group on $n$ letters. Every finite group is isomorphic to a subgroup of $S_n$ for some $n \in \Z$.
Coprimality Relation is not Antisymmetric
:$\perp$ is not antisymmetric.
Coprimality Relation is Non-Transitive
:$\perp$ is non-transitive.
Canonical Injection into Cartesian Product of Modules
Let $G$ be the cartesian product of a sequence $\sequence {G_n}$ of $R$-modules. Then for each $j \in \closedint 1 n$, the canonical injection $\inj_j$ from $G_j$ into $G$ is a monomorphism.
Gaussian Integers does not form Subfield of Complex Numbers
The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ is not a subfield of $\C$.
Even Integers form Commutative Ring
Let $2 \Z$ be the set of even integers. Then $\struct {2 \Z, +, \times}$ is a commutative ring. However, $\struct {2 \Z, +, \times}$ is not an integral domain.
Leibniz's Rule/One Variable/Second Derivative
Let $f$ and $g$ be real functions defined on the open interval $I$. Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are twice differentiable. Then: :$\paren {\map f x \, \map g x}'' = \map f x \, \map {g''} x + 2 \map {f'} x \, \map {g'} x + \map {f''} x \, \map g x$
Vector Space on Cartesian Product is Vector Space
Let $\struct {K, +, \circ}$ be a division ring. Let $n \in \N_{>0}$. Let $\struct {K^n, +, \times}_K$ be the '''$K$-vector space $K^n$'''. Then $\struct {K^n, +, \times}_K$ is a $K$-vector space.
Ring of Linear Operators
Let $\map {\LL_R} G$ be the set of all linear operators on $G$. {{explain|the precise nature of $G$}} Let $\phi \circ \psi$ denote the composition of the two linear operators $\phi$ and $\psi$. Then $\struct {\map {\LL_R} G, +, \circ}$ is a ring.
No Non-Trivial Norm on Rational Numbers is Complete
No non-trivial norm on the set of the rational numbers is complete.
Inverse Evaluation Isomorphism of Annihilator
Let $R$ be a commutative ring.  Let $G$ be a module over $R$ whose dimension is finite. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $N$ be a submodule of $G^*$. Let $J$ be the evaluation isomorphism from $G$ onto $G^{**}$. Let $N^\circ$ be the annihilator of $N$. Then: :$J^{-1} \left({N^\circ}\right) = \left\{{x \in G: \forall t' \in N: t' \left({x}\right) = 0}\right\}$
Complex Numbers form Vector Space over Reals
Let $\R$ be the set of real numbers. Let $\C$ be the set of complex numbers. Then the $\R$-module $\C$ is a vector space.
Dimension of Vector Space on Cartesian Product
Let $\struct {K, +, \circ}$ be a division ring. Let $n \in \N_{>0}$. Let $\mathbf V := \struct {K^n, +, \times}_K$ be the '''$K$-vector space $K^n$'''. Then the dimension of $\mathbf V$ is $n$.
Finitely Generated Vector Space has Basis
Let $K$ be a division ring. Let $V$ be a finitely generated vector space over $K$. Then $V$ has a finite basis.
Primitive of Hyperbolic Cosecant Function/Logarithm Form
:$\displaystyle \int \csch x \rd x = -\ln \size {\csch x + \coth x} + C$ where $\csch x + \coth x \ne 0$.
Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form
:$\displaystyle \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$
Primitive of Product of Hyperbolic Secant and Tangent
:$\ds \int \sech x \tanh x \rd x = -\sech x + C$ where $C$ is an arbitrary constant.
Existence of Subgroup whose Index is Prime Power
Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ which has a finite index in $G$. Let: :$p^k \divides \index G H$ where: :$p$ is a prime number :$k \in \Z_{>0}$ is a (strictly) positive integer :$\divides$ denotes divisibility. Then $G$ contains a subgroup $K$ such that: :$\index K H = p^k$
Condition for Planes to be Parallel
Let $P: \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma$ be a plane in $\R^3$. Then the plane $P'$ is parallel to $P$ iff there is a $\gamma' \in \R$ such that: :$P' = \left\{{ \left({x_1, x_2, x_3}\right) \in \R^3 : \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3 = \gamma' }\right\}$
Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2
:$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$
Invertible Matrix Corresponds with Change of Basis
Let $R$ be a commutative ring with unity. Let $G$ be an $n$-dimensional unitary $R$-module. Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$. Let $\mathbf P = \left[{\alpha}\right]_{n}$ be a square matrix of order $n$ over $R$. Let $\displaystyle \forall j \in \left[{1 \,.\,.\, n}\right]: b_j = \sum_{i \mathop = 1}^n \alpha_{i j} a_i$. Then $\left \langle {b_n} \right \rangle$ is an ordered basis of $G$ iff $\mathbf P$ is invertible.
Matrix Similarity is Equivalence Relation
Matrix similarity is an equivalence relation.
Similar Matrices are Equivalent
If two square matrices over a ring with unity $R$ are similar, then they are equivalent. It follows directly that every equivalence class for the relation of similarity on $\mathcal M_R \left({n}\right)$ is contained in an equivalence class for the relation of matrix equivalence. Here, $\mathcal M_R \left({n}\right)$ denotes the $n \times n$ matrix space over $R$.
Rank is Dimension of Subspace
Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Then the rank of $\mathbf A$ is the dimension of the subspace of $K^n$ generated by the rows of $\mathbf A$.