Irrational Numbers form G-Delta Set in Reals Let $\R \setminus \Q$ denote the set of irrational numbers. Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then $\R \setminus \Q$ forms a $G_\delta$ set in $\R$.


Automorphism Maps Generator to Generator Let $G$ be a cyclic group. Let $g$ be a generator of $G$. Let $\phi$ be an automorphism on $G$. Then $\map \phi g$ is also a generator of $G$.


Hermitian Matrix has Real Eigenvalues Every Hermitian matrix has eigenvalues which are all real numbers.


Fortissimo Space is not Weakly Countably Compact Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not weakly countably compact.


Equivalent Statements for Congruence Modulo Subgroup/Right Let $x \equiv^r y \pmod H$ denote that $x$ is right congruent modulo $H$ to $y$. Then the following statements are equivalent: {{begin-eqn}} {{eqn | n = 1       | l = x       | o = \equiv^r       | r = y \pmod H }} {{eqn | n = 2       | l = x y^{-1}       | o = \in       | r = H }} {{eqn | n = 3       | l = \exists h \in H: x y^{-1}       | r = h }} {{eqn | n = 4       | l = \exists h \in H: x       | r = h y }} {{end-eqn}}


Existence-Uniqueness Theorem for Homogeneous First-Order Differential Equation Let $P \left({x}\right)$ be a continuous function on an open interval $I \subseteq \R$. Let $a \in I$. Let $b \in \R$. Let $f \left({x}\right) = y$ be a function satisfying the differential equation: :$y' + P \left({x}\right) y = 0$ and the initial condition: :$f \left({a}\right) = b$ Then there exists a unique function satisfying these initial conditions on the interval $I$. That function takes the form: :$f \left({x}\right) = b e^{-A \left({x}\right)}$ where: :$\displaystyle A \left({x}\right) = -\int_a^x P \left({t}\right) \mathrm d t$


Non-Zero Integers are Cancellable for Multiplication Every non-zero integer is cancellable for multiplication. That is: :$\forall x, y, z \in \Z, x \ne 0: x y = x z \iff y = z$


Linearly Independent Solutions of y'' - y = 0 The second order ODE: :$(1): \quad y'' - y = 0$ has solutions: :$y_1 = e^x$ :$y_2 = e^{-x}$ which are linearly independent.


Characterization of Prime Filter by Finite Suprema Let $L = \struct {S, \vee, \preceq}$ be a join semilattice. Let $F$ be a filter in $L$. Then :$F$ is a prime filter {{iff}}: :for all non-empty finite subset $A$ of $S: \paren {\sup A \in F \implies \exists a \in A: a \in F}$


Upper Semilattice on Classical Set is Semilattice Let $\struct {S, \vee}$ be an upper semilattice on a classical set $S$. Then $\struct {S, \vee}$ is a semilattice.


Set of Division Subrings forms Complete Lattice Let $\struct {D, +, \circ}$ be a division ring. Let $\mathbb K$ be the set of all division subrings of $K$. Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.


Primitive of x over 1 plus Cosine of a x :$\displaystyle \int \frac {x \rd x} {1 + \cos a x} = \frac x a \tan \frac {a x} 2 + \frac 2 {a^2} \ln size {\cos \frac {a x} 2} + C$


Principle of Stationary Action with Standard Lagrangian implies Newton's Laws of Motion Let $\MM$ be an $n$-dimensional Euclidean manifold. Let $P$ be a physical system composed of a countable number of classical particles with inertial masses $m_i$ with $i \in \N$. Let $\mathbf x = \map {\mathbf x} t$ be twice-differentiable vector-valued function embedded in $\MM$. Suppose ${\mathbf x}_i$ represents the position of the $i$-th particle of $P$. Suppose the action of $P$ is of the following form: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian. Suppose, all (internal or external) forces ${\mathbf F}_i$ acting upon $P$ are of the form: :${\mathbf F}_i = - \dfrac {\partial U} {\partial \mathbf x_i}$ where: :$\displaystyle U = \map U {t, \set{\mathbf x_i} }$ is a differentiable real function :$\set {\mathbf x_i}$ denotes dependence on the positions of all the particles. Then the stationary point of $S$ implies Newton's Second Law of Motion.


Addition of Fractions Let $a, b, c, d \in \Z$ such that $b d \ne 0$. Then: :$\dfrac a b + \dfrac c d = \dfrac {a D + B c} {\lcm \set {b, d} }$ where: :$B = \dfrac b {\gcd \set {b, d} }$ :$D = \dfrac d {\gcd \set {b, d} }$ :$\lcm$ denotes lowest common multiple :$\gcd$ denotes greatest common divisor.


Median of Exponential Distribution Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$.  Then the median of $X$ is equal to $\beta \ln 2$.


Minus One is Less than Zero :$-1 < 0$


Equivalence of Definitions of Semantic Equivalence for Predicate Logic Let $\mathbf A, \mathbf B$ be WFFs of predicate logic. {{TFAE|def = Semantic Equivalence (Predicate Logic)|semantic equivalence}}


Sums of Partial Sequences of Squares Let $n \in \Z_{>0}$. Consider the odd number $2 n + 1$ and its square $\paren {2 n + 1}^2 = 2 m + 1$. Then: :$\displaystyle \sum_{j \mathop = 0}^n \paren {m - j}^2 = \sum_{j \mathop = 1}^n \paren {m + j}^2$ That is: :the sum of the squares of the $n + 1$ integers up to $m$ equals: :the sum of the squares of the $n$ integers from $m + 1$ upwards.


Mapping is Bijection iff Composite with Direct Image Mapping with Complementation Commutes Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: :$f$ is a bijection {{iff}}: :$f^\to \circ \complement_S = \complement_T \circ f^\to$ where: :$f^\to: \powerset S \to \powerset T$ denotes the direct image mapping of $f$ :$\complement_S: \powerset S \to \powerset S$ denotes the complement relative to $S$ :$\complement_T: \powerset T \to \powerset T$ denotes the complement relative to $T$ :$\powerset S$ and $\powerset T$ denote the power sets of $S$ and $T$ respectively.


Interior of Intersection may not equal Intersection of Interiors Let $T$ be a topological space. Let $\mathbb H$ be a set of subsets of $T$. That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then it is not necessarily the case that: :$\displaystyle \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ = \bigcap_{H \mathop \in \mathbb H} H^\circ$ where $H^\circ$ denotes the interior of $H$.


Excess Kurtosis of Chi-Squared Distribution Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac {12} n$


Set together with Omega-Accumulation Points is not necessarily Closed Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $\Omega$ denote the set of $\omega$-accumulation points of $H$. Then it is not necessarily the case that $H \cup \Omega$ is a closed set of $T$.


Equivalence of Definitions of Convergent Sequence in Metric Space 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}}


Laplace Transform of Generating Function of Sequence Let $\left\langle{a_n}\right\rangle$ be a sequence which has a generating function which is convergent. Let $G \left({z}\right)$ be the generating function for $\left\langle{a_n}\right\rangle$. Let $f \left({x}\right)$ be the step function: :$f \left({x}\right) = \displaystyle \sum_{k \mathop \in \Z} a_k \left[{0 \le k \le x}\right]$ where $\left[{0 \le k \le x}\right]$ is Iverson's convention. Then the Laplace transform of $f \left({x}\right)$ is given by: :$\mathcal L \left\{ {f \left({s}\right)}\right\} = \dfrac {G \left({e^{-s} }\right)} s$


Existence of Local Coordinates Let $X^n$ be an $n$-dimensional manifold. Let $U$ be a neighborhood of a point $p \in X^n$. Then there exist local coordinates on $U$.


Laplace Transform of Half Wave Rectified Sine Curve Consider the half wave rectified sine curve: :$\map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$ The Laplace transform of $\map f t$ is given by: :$\laptrans {\map f t} = \dfrac 1 {\paren {1 - e^{-\pi s} } \paren {s^2 + 1} }$


Primitive of Arccosecant of x over a :$\displaystyle \int \operatorname{arccsc} \frac x a \rd x = \begin{cases} \displaystyle x \operatorname{arccsc} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \displaystyle x \operatorname{arccsc} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$


Dedekind-Complete Bounded Ordered Set is Complete Lattice Let $\left({L, \preceq}\right)$ be an ordered set. Let $L$ have a lower bound $\bot$ and an upper bound $\top$. Let $\left({L, \preceq}\right)$ be Dedekind-complete. Then $\left({L, \preceq}\right)$ is a complete lattice.


Linear Second Order ODE/y'' - 2 y' + 5 y = 25 x^2 + 12 The second order ODE: :$(1): \quad y'' - 2 y' + 5 y = 25 x^2 + 12$ has the general solution: :$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x} + 2 + 4 x + 5 x^2$


Non-Equivalence as Conjunction of Disjunction with Disjunction of Negations :$\neg \paren {p \iff q} \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$


Increasing Sum of Binomial Coefficients :$\displaystyle \sum_{j \mathop = 0}^n j \binom n j = n 2^{n - 1}$


Equivalence of Axiom Schemata for Groups/Warning Suppose we build an algebraic structure with the following axioms: {{begin-axiom}} {{axiom | n = 0         | lc= Closure Axiom         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = 1         | lc= Associativity Axiom         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = 2         | lc= Right Identity Axiom         | q = \exists e \in G: \forall a \in G         | m = a \circ e = a }} {{axiom | n = 3         | lc= Left Inverse Axiom         | q = \forall x \in G: \exists b \in G         | m = b \circ a = e }} {{end-axiom}} Then this does '''not''' (necessarily) define a group (although clearly a group fulfils those axioms).


Laplace Transform of Dirac Delta Function by Function Let $\map f t: \R \to \R$ or $\R \to \C$ be a function. Let $\map \delta t$ denote the Dirac delta function. Let $c$ be a positive constant real number. Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Then: :$\laptrans {\map \delta {t - c} \, \map f t} = e^{- s c} \, \map f c$


Generating Function for Binomial Coefficients Let $\left \langle {a_n}\right \rangle$ be the sequence defined as: : $\displaystyle \forall n \in \N: a_n = \begin{cases} \binom m n & : n = 0, 1, 2, \ldots, m \\ 0 & : \text{otherwise}\end{cases}$ where $\displaystyle \binom m n$ denotes a binomial coefficient. Then the generating function for $\left \langle {a_n}\right \rangle$ is given as: : $\displaystyle G \left({z}\right) = \sum_{n \mathop = 0}^m \binom m n z^n = \left({1 + z}\right)^m$


Equivalence of Definitions of Second Chebyshev Function {{TFAE|def = Second Chebyshev Function}} :$(1): \quad \displaystyle \map \psi x = \sum_{p^k \mathop \le x} \ln p$ :$(2): \quad \displaystyle \map \psi x = \sum_{1 \mathop \le n \mathop \le x} \map \Lambda n$ :$(3): \quad \displaystyle \map \psi x = \sum_{p \mathop \le x} \floor {\log_p x} \ln p$ where: :$p$ is a prime number :$\map \Lambda n$ is the von Mangoldt function :$\floor {\, \cdot \, }$ denotes the floor function.


3-Digit Numbers forming Longest Reverse-and-Add Sequence Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation. Let $r \left({m}\right)$ be the reverse-and-add process on $m$. Let $r$ be applied iteratively to $m$. The $3$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are: :$187, 286, 385, 583, 682, 781, 880$ all of which need $23$ iterations. :$869$ and $968$ are the result of the first iteration for each of these, and so take $22$ iterations to reach a palindromic number.


Rational Number Space is Sigma-Compact Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is $\sigma$-compact.


Smallest Numbers with 240 Divisors The smallest integers with $240$ divisors are: :$720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$


Signed Stirling Number of the First Kind of 0 :$s \left({0, n}\right) = \delta_{0 n}$


Primitive of x cubed by Root of x squared plus a squared :$\displaystyle \int x^3 \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^3} 3 + C$


L1 Metric on Closed Real Interval is Metric Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let $d: S \times S \to \R$ be the $L^1$ metric on $\left[{a \,.\,.\, b}\right]$: :$\displaystyle \forall f, g \in S: d \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$ Then $d$ is a metric.


Ambiguous Case for Spherical Triangle Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the sides $a$ and $b$ be known. Let the angle $\sphericalangle B$ also be known. Then it may not be possible to know the value of $\sphericalangle A$. This is known as the '''ambiguous case (for the spherical triangle)'''.


Double of Antiperiod is Period Let $f: \R \to \R$ be a real antiperiodic function with an anti-period of $A$. Then $f$ is also periodic with a period of $2A$.


Set of Strictly Positive Real Numbers has no Smallest Element Let $\R_{>0}$ denote the set of strictly positive real numbers. Then $\R_{>0}$ has no smallest element.


Subgroup of Order 1 is Trivial Let $\struct {G, \circ}$ be a group. Then $\struct {G, \circ}$ has exactly $1$ subgroup of order $1$: the trivial subgroup.


Number of Odd Entries in Row of Pascal's Triangle is Power of 2 The number of odd entries in a row of Pascal's triangle is a power of $2$.


Sextuple Angle Formula for Cosine : $\cos \paren {6 \theta} = 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta - 1$


Irrational Number Space is Paracompact Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is paracompact.


Existence of Infinitely Many Integrating Factors Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be such that $M$ and $N$ are real functions of two variables which are ''not'' homogeneous functions of the same degree. Suppose $(1)$ has an integrating factor. Then $(1)$ has an infinite number of integrating factors


Ring Monomorphism from Integers to Rationals Let $\phi: \Z \to \Q$ be the mapping from the integers $\Z$ to the rational numbers $\Q$ defined as: :$\forall x \in \Z: \map \phi x = \dfrac x 1$ Then $\phi$ is a (ring) monomorphism, but specifically not an epimorphism.


Generator for Almost Isosceles Pythagorean Triangle Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T = \tuple {a, b, c}$. Let the generator for $T$ be $\tuple {m, n}$. Then: :$P$ is almost isosceles {{iff}} :$\tuple {2 m + n, m}$ is the generator for the Pythagorean triple $T'$ of another almost isosceles Pythagorean triangle $P'$.


Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors The following sequence of integers have the property that both the harmonic mean and arithmetic mean of their divisors are integers: :$1, 6, 140, 270, 672, \ldots$ {{OEIS|A007340}}


ODE/(D^4 - 1) y = sin x The second order ODE: :$(1): \quad \paren {D^4 - 1} y' = \sin x$ has a general solution: :$y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$


Value of b for b by Logarithm Base b of x to be Minimum Let $x \in \R_{> 0}$ be a (strictly) positive real number. Consider the real function $f: \R_{> 0} \to \R$ defined as: :$f \left({b}\right) := b \log_b x$ $f$ attains a minimum when :$b = e$ where $e$ is Euler's number.


Order of Automorphism Group of Prime Group Let $p$ be a prime number. Let $G$ be a group of order $p$. Let $\Aut G$ denote the automorphism group of $G$. Then: :$\order {\Aut G} = p - 1$ where $\order {\, \cdot \,}$ denotes the order of a group.


Smallest Odd Number not of form 2 a squared plus p $17$ is the smallest odd number $n$ greater than $3$ which cannot be expressed in the form: :$n = 2 a^2 + p$ where: :$p$ is prime :$a \in \Z_{>0}$ is a (strictly) positive integer.


Complex Numbers under Multiplication do not form Group The algebraic structure $\struct {\C, \times}$ consisting of the set of complex numbers $\C$ under multiplication $\times$ is not a group.


Numbers of Type Rational a plus b root 2 form Field/Corollary The field $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ is a subfield of $\struct {\R, +, \times}$.


Pi Squared is Irrational Pi squared ($\pi^2$) is irrational.


Bessel Function of the First Kind of Negative Integer Order :$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$


Derivative of Arcsecant Function/Corollary 2 :$\dfrac {\mathrm d \left({\operatorname{arcsec} x }\right)} {\mathrm d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2}}}$


Multiplication of Cross-Relation Equivalence Classes on Natural Numbers is Well-Defined Let $\struct {\N, +}$ be the semigroup of natural numbers under addition. Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$. Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ Let $\eqclass {x, y} {}$ denote the equivalence class of $\tuple {x, y}$ under $\boxtimes$. Let $\otimes$ be the binary operation defined on these equivalence classes as: :$\forall \eqclass {a, b} {}, \eqclass {c, d} {} \in \N \times \N: \eqclass {a, b} {} \otimes \eqclass {c, d} {} = \eqclass {\tuple {a \cdot c} + \tuple {b \cdot d}, \tuple {a \cdot d} + \tuple {b \cdot c} } {}$ where $a \cdot c$ denotes natural number multiplication between $a$ and $c$. {{WIP|Introduce the language of the Definition:Quotient Set.}} The operation $\otimes$ on these equivalence classes is well-defined, in the sense that: {{begin-eqn}} {{eqn | l = \eqclass {a_1, b_1} {}       | r = \eqclass {a_2, b_2} {}       | c =  }} {{eqn | l = \eqclass {c_1, d_1} {}       | r = \eqclass {c_2, d_2} {}       | c =  }} {{eqn | ll= \leadsto       | l = \eqclass {a_1, b_1} {} \otimes \eqclass {c_1, d_1} {}       | r = \eqclass {a_2, b_2} {} \otimes \eqclass {c_2, d_2} {}       | c =  }} {{end-eqn}}


Excluded Point Topology is T4 Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space. Then $T$ is a $T_4$ space.


De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Corollary :$T_1 \cup T_2 = \overline {\overline T_1 \cap \overline T_2}$


Determinant with Columns Transposed If two columns of a matrix with determinant $D$ are transposed, its determinant becomes $-D$.


Summation of Powers over Product of Differences/Example {{begin-eqn}} {{eqn | l = \frac 1 {\paren {a - b} \paren {a - c} } + \frac 1 {\paren {b - a} \paren {b - c} } + \frac 1 {\paren {c - a} \paren {c - b} }       | r = 0 }} {{eqn | l = \frac a {\paren {a - b} \paren {a - c} } + \frac b {\paren {b - a} \paren {b - c} } + \frac c {\paren {c - a} \paren {c - b} }       | r = 0 }} {{eqn | l = \frac {a^2} {\paren {a - b} \paren {a - c} } + \frac {b^2} {\paren {b - a} \paren {b - c} } + \frac {c^2} {\paren {c - a} \paren {c - b} }       | r = 1 }} {{eqn | l = \frac {a^3} {\paren {a - b} \paren {a - c} } + \frac {b^3} {\paren {b - a} \paren {b - c} } + \frac {c^3} {\paren {c - a} \paren {c - b} }       | r = a + b + c }} {{end-eqn}}


Z-Module Associated with Abelian Group is Unitary Z-Module Let $\struct {G, *}$ be an abelian group with identity $e$. Let $\struct {G, *, \circ}_\Z$ be the $Z$-module associated with $G$. Then $\struct {G, *, \circ}_\Z$ is a unitary $Z$-module.


Equivalence of Definitions of Nilradical of Ring {{TFAE|def = Nilradical of Ring}} Let $A$ be a commutative ring.


Multiplicative Group of Reduced Residues Modulo 7 is Cyclic Let $\struct {\Z'_7, \times_7}$ denote the multiplicative group of reduced residues modulo $7$. Then $\struct {\Z'_7, \times_7}$ is cyclic.


Multiplicative Group of Rationals is Subgroup of Complex Let $\struct {\Q, \times}$ be the multiplicative group of rational numbers. Let $\struct {\C, \times}$ be the multiplicative group of complex numbers. Then $\struct {\Q, \times}$ is a normal subgroup of $\struct {\C, \times}$.


Trace of Unit Matrix Let $\mathbf I_n$ be the unit matrix of order $n$. Then: :$\map \tr {\mathbf I_n} = n$ where $\map \tr {\mathbf I_n}$ denotes the trace of $\mathbf I_n$.


Even Order Group has Order 2 Element Let $G$ be a group whose identity is $e$. Let $G$ be of even order. Then: :$\exists x \in G: \order x = 2$ That is: :$\exists x \in G: x \ne e: x^2 = e$


Power Series Expansion for Logarithm of Sine of x {{begin-eqn}} {{eqn | l = \ln \size {\sin x}       | r = \ln \size x - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}       | c =  }} {{eqn | r = \ln \size x - \frac {x^2} 6 - \frac {x^4} {180} - \frac {x^6} {2835} + \cdots       | c =  }} {{end-eqn}} for all $x \in \R$ such that $0 < \size x < \pi$.


Equal Powers of Group Element implies Finite Order Let $\struct {G, \circ}$ be a group. Let $g \in G$ such that $g^r = g^s$ where $r, s \in \Z: r \ne s$. Then there exists $m \in \Z_{>0}$ such that: :$(1): \quad g^m = e$ :$(2): \quad 0 \le i < j < m \implies g^i \ne g^j$


Free Commutative Monoid is Commutative Monoid The free commutative monoid on a set $\family {X_j: j \in J}$ is a commutative monoid.


Complement of Open Set in Complex Plane is Closed Let $S \subseteq \C$ be an open subset of the complex plane $\C$. Then the complement of $S$ in $\C$ is closed.


Quotient Subgroup of Semigroup Induced on Power Set Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a normal subgroup of $\struct {G, \circ}$. Then $\struct {G / H, \circ_H}$ is a subgroup of $\struct {\powerset G, \circ_\mathcal P}$, where: :$\struct {G / H, \circ_H}$ is the quotient group of $G$ by $H$ :$\struct {\powerset G, \circ_\mathcal P}$ is the semigroup induced by the operation $\circ$ on the power set $\powerset G$ of $G$.


Quotient Mapping of Inverse Completion Let $\struct {T, \circ'}$ be an inverse completion of a commutative semigroup $\struct {S, \circ}$, where $C$ is the set of cancellable elements of $S$. Let $f: S \times C: T$ be the mapping defined as: :$\forall x \in S, y \in C: \map f {x, y} = x \circ' y^{-1}$ Let $\RR_f$ be the equivalence relation induced by $f$. Then: :$\tuple {x_1, y_1} \mathrel {\RR_f} \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$


Beta Function is Defined for Positive Reals Let $x, y \in \R$ be real numbers. Let $\Beta \left({x, y}\right)$ be the Beta function: :$\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$ Then $\Beta \left({x, y}\right)$ exists provided that $x, y > 0$.


Continuous implies Increasing in Scott Topological Lattices Let $T_1 = \struct {S_1, \preceq_1, \tau_1}$ and $T_2 = \struct {S_2, \preceq_2, \tau_2}$ be up-complete topological lattices with Scott topologies. Let $f: S_1 \to S_2$ be a continuous mapping. Then $f$ is an increasing mapping.


Powers of Group Elements/Sum of Indices/Additive Notation :$\forall m, n \in \Z: m g + n g = \paren {m + n} g$


Automorphism Group of Cyclic Group is Abelian Let $G$ be a cyclic group. Let $\Aut G$ denote the automorphism group of $G$. Then $\Aut G$ is abelian.


Direct Image of Intersection with Inverse Image Then: :$\forall A \in \powerset S, B \in \powerset T: \map {f^\to} {A \cap \map {f^\gets} B} = \map {f^\to} A \cap B$


Odd Integers under Multiplication do not form Group Let $S$ be the set of odd integers: :$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$ Let $\struct {S, \times}$ denote the algebraic structure formed by $S$ under the operation of multiplication. Then $\struct {S, \times}$ is not a group.


Identity Mapping is Idempotent Let $S$ be a set. Let $I_S: S \to S$ be the identity mapping on $S$. Then $I_S$ is idempotent: :$I_S \circ I_S = I_S$


Primitive of Reciprocal of Sine of a x by 1 minus Cosine of a x :$\displaystyle \int \frac {\mathrm d x} {\sin a x \left({1 - \cos a x}\right)} = \frac {-1} {2 a \left({1 - \cos a x}\right)} + \frac 1 {2 a} \ln \left\vert{\tan \frac {a x} 2}\right\vert + C$


Difference of Two Fourth Powers :$x^4 - y^4 = \paren {x - y} \paren {x + y} \paren {x^2 + y^2}$


Linear Second Order ODE/y'' - 2 y' + y = 1 over 1 + e^x The second order ODE: :$(1): \quad y'' - 2 y' + y = \dfrac 1 {1 + e^x}$ has a particular solution: :$y = 1 + e^x \displaystyle \int \map \ln {1 + e^{-x} } \rd x$


Power Series Expansion for General Exponential Function Let $a \in \R_{> 0}$ be a (strictly) positive real number. Then: Then: {{begin-eqn}} {{eqn | ll= \forall x \in \R:       | l = a^x       | r = \sum_{n \mathop = 0}^\infty \frac {\left({x \ln a}\right)^n} {n!}       | c =  }} {{eqn | r = 1 + x \ln a + \frac {\left({x \ln a}\right)^2} {2!} + \frac {\left({x \ln a}\right)^3} {3!} + \cdots       | c =  }} {{end-eqn}}


Polynomials Closed under Addition/Polynomial Forms Let: : $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ : $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over the ring $R$. Then the operation of polynomial addition on $f$ and $g$: Define the sum: :$\displaystyle f \oplus g = \sum_{k \mathop \in Z} \left({a_k + b_k}\right) \mathbf X^k$ Then $f \oplus g$ is a polynomial. That is, the operation of polynomial addition is closed on the set of all polynomials on a given set of indeterminates $\left\{{X_j: j \in J}\right\}$.


Coprimality Relation is Symmetric :$\perp$ is symmetric.


Bound for Difference of Irrational Number with Convergent Let $x$ be an irrational number. Let $\sequence {C_n}$ be the sequence of convergents of the continued fraction expansion of $x$. Then $\forall n \ge 1$: :$C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$ :$\size {x - C_n} < \dfrac 1 {q_n q_{n + 1} }$


Infinite Product of Weakly Locally Compact Spaces is not always Weakly Locally Compact Let $I$ be an indexing set with infinite 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 weakly locally compact-compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also weakly locally compact.


Equivalence of Definitions of Continuous Mapping between Topological Spaces/Point Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$. Let $x \in S_1$. {{TFAE|def = Continuous at Point of Topological Space|view = continuity at a point of a topological space}} === Definition using Open Sets === {{Definition:Continuous Mapping (Topology)/Point/Open Sets}} === Definition using Filters === {{Definition:Continuous Mapping (Topology)/Point/Filters}}


Dual of Total Ordering is Total Ordering Let $\preccurlyeq$ be a total ordering. Then its dual ordering $\succcurlyeq$ is also a total ordering.


Ordered Group Equivalences Let $\left({S, \circ, \preceq}\right)$ be an ordered group whose identity is $e$. Let $x, y, z, \in S$. Then the following are all equivalent: : $(1): \quad x \prec y$ : $(2): \quad x \circ z \prec y \circ z$ : $(3): \quad z \circ x \prec z \circ y$ : $(4): \quad y^{-1} \prec x^{-1}$ : $(5): \quad e \prec y \circ x^{-1}$ : $(6): \quad e \prec x^{-1} \circ y$


Integration by Inversion :$\displaystyle \int_0^{+\infty} \map f x \rd x = \int_0^{+\infty} \dfrac {\map f {\frac 1 x} } {x^2} \rd x$


Compact Hausdorff Space is Locally Compact Let $T = \left({S, \tau}\right)$ be a Hausdorff space which is compact. Then $T$ is locally compact.


Real Addition is Well-Defined The operation of addition on the set of real numbers $\R$ is well-defined.


Power Function is Completely Multiplicative/Integers Let $c \in \Z$ be an integer. Let $f_c: \Z \to \Z$ be the mapping defined as: :$\forall n \in \Z: \map {f_c} n = n^c$ Then $f_c$ is completely multiplicative.


Ceiling Function/Examples/Ceiling of Minus One Half :$\ceiling {- \dfrac 1 2} = 0$


Primitive of x by Inverse Hyperbolic Sine of x over a :$\displaystyle \int x \sinh^{-1} \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \sinh^{-1} \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$


Categories of Elements of Ring Let $\left({R, +, \circ}\right)$ be a ring. The elements of $R$ are partitioned into three classes: :$(1): \quad$ the zero :$(2): \quad$ the units :$(3): \quad$ the proper elements.


Permutation of Indices Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers. Let the fiber of truth of $R$ be finite. Let  $\pi$ is a permutation on the fiber of truth of $R$. Then:


Primitive of Half Integer Power of a x squared plus b x plus c Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x = \frac {\left({2 a x + b}\right) \left({a x^2 + b x + c}\right)^{n + \frac 1 2} } {4 a \left({n + 1}\right)} + \frac {\left({2 n + 1}\right) \left({4 a c - b^2}\right)} {8 a  \left({n + 1}\right)} \int \left({a x^2 + b x + c}\right)^{n - \frac 1 2} \ \mathrm d x$


Rule of Association/Disjunction/Formulation 2/Forward Implication :$\vdash \left({p \lor \left({q \lor r}\right)}\right) \implies \left({\left({p \lor q}\right) \lor r}\right)$


Sum of Squares of Divisors of 24 and 26 are Equal The sum of the squares of the divisors of $24$ equals the sum of the squares of the divisors of $26$: :$\map {\sigma_2} {24} = \map {\sigma_2} {26}$ where $\sigma_\alpha$ denotes the divisor function.


Successor Mapping on Natural Numbers is not Surjection Let $f: \N \to \N$ be the successor mapping on the natural numbers $\N$: :$\forall n \in \N: \map f n = n + 1$ Then $f$ is not a surjection.


Sum of Fourth Powers with Product of Squares :$x^4 + x^2 y^2 + y^4 = \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$


Primitive of Arcsecant of x over a/Formulation 2 :$\displaystyle \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$ for $x^2 > 1$. $\displaystyle \arcsec \frac x a$ is undefined on the real numbers for $x^2 < 1$.


Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Cosine :$\displaystyle \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C$


Magnitude and Direction of Equilibrant Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space. The '''equilibrant''' $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ is: :$\mathbf E = -\displaystyle \sum_{k \mathop = 1}^n \mathbf F_k$ That is, the magnitude and direction of $\mathbf E$ is such as to balance out the effect of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$.


Finite Union of Closed Sets is Closed/Normed Vector Space Let $M = \struct {X, \norm {\, \cdot \, }}$ be a normed vector space. Then the union of finitely many closed sets of $M$ is itself closed.


Set of Integers is not Well-Ordered by Usual Ordering The set of integers $\Z$ is not well-ordered under the usual ordering $\le$.


Terminal Velocity of Body under Fall Retarded Proportional to Square of Velocity Let $B$ be a body falling in a gravitational field. Let $B$ be falling through a medium which exerts a resisting force of magnitude $k v^2$ upon $B$ which is proportional to the square of the velocity of $B$ relative to the medium. Then the terminal velocity of $B$ is given by: :$v =  \sqrt {\dfrac {g m} k}$


Poset Elements Equal iff Equal Weak Lower Closure Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $s, t \in S$. Then $s = t$ {{iff}}: :$s^\preccurlyeq = t^\preccurlyeq$ where $s^\preccurlyeq$ denotes weak lower closure of $s$. That is, {{iff}}, for all $r \in S$: :$r \preccurlyeq s \iff r \preccurlyeq t$


Existence of Prime-Free Sequence of Natural Numbers Let $n$ be a natural number. Then there exists a sequence of consecutive natural numbers of length $n$ which are all composite.


Left and Right Operation Closed for All Subsets Let $S$ be a set. Let: : $\leftarrow$ be the left operation on $S$ : $\rightarrow$ be the right operation on $S$. That is: : $\forall x, y \in S: x \leftarrow y = x$ : $\forall x, y \in S: x \rightarrow y = y$ Let $\mathcal P \left({S}\right)$ be the power set of $S$. Then for all $T \in \mathcal P \left({S}\right)$, both $\leftarrow$ and $\rightarrow$ are closed on $T$. Thus, for all $T \in \mathcal P \left({S}\right)$: : $\left({T, \leftarrow}\right)$ is a subsemigroup of $\left({S, \leftarrow}\right)$ : $\left({T, \rightarrow}\right)$  is a subsemigroup of $\left({S, \rightarrow}\right)$.


Coproduct of Free Monoids Let $\mathbf {Mon}$ be the category of monoids. Let $\map M A$ and $\map M B$ be free monoids on sets $A$ and $B$, respectively. Let $A \sqcup B$ be the disjoint union of $A$ and $B$. Then the free monoid $\map M {A \sqcup B}$ on $A \sqcup B$ is the coproduct of $\map M A$ and $\map M B$ in $\mathbf {Mon}$.


Sum of Elements in Inverse of Vandermonde Matrix Let $V_n$ be the Vandermonde matrix of order $n$ given by: :$V_n = \begin{bmatrix}   x_1 & x_2 & \cdots & x_n \\   x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\   x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$ Let $V_n^{-1}$ be its inverse, from Inverse of Vandermonde Matrix: :$b_{i j} = \begin{cases} \paren {-1}^{j - 1} \paren {\dfrac{\displaystyle \sum_{\substack {1 \mathop \le m_1 \mathop < \ldots \mathop < m_{n - j} \mathop \le n \\ m_1, \ldots, m_{n - j} \mathop \ne i} } x_{m_1} \cdots x_{m_{n - j} } } {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_m - x_i} } } & : 1 \le  j < n \\ \qquad \qquad \qquad \dfrac 1 {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_i - x_m} } & : j = n \end{cases}$ The sum of all the elements of $V_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = 1 - \prod_{k \mathop = 1}^n \paren {1 - \dfrac 1 {x_k} }$


Intersection 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: :$\map \alpha {A \cap B} = \map \alpha A \cap \map \alpha B$


Existence of Non-Standard Models of Arithmetic There exist non-standard models of arithmetic.


Beta Function of Real Number with 1 :$\Beta \left({x, 1}\right) = \Beta \left({1, x}\right) = \dfrac 1 x$


Group Acts on Itself Let $\struct {G, \circ}$ be a group whose identity is $e$. Then $\struct {G, \circ}$ acts on itself by the rule: :$\forall g, h \in G: g * h = g \circ h$


Characteristic Function of Union/Variant 1 :$\chi_{A \mathop \cup B} = \min \set {\chi_A + \chi_B, 1}$


Pandigital Sum whose Components are Multiples The following sums use each of the digits $1$ to $9$ exactly once each, while one summand equals $2$ times the other summand: {{begin-eqn}} {{eqn | l = 192 + 384       | r = 576       | c = where $2 \times 192 = 384$ and so $3 \times 192 = 576$ }} {{eqn | l = 219 + 438       | r = 657       | c = where $2 \times 219 = 438$ and so $3 \times 219 = 657$ }} {{eqn | l = 273 + 546       | r = 819       | c = where $2 \times 273 = 546$ and so $3 \times 273 = 819$ }} {{eqn | l = 327 + 654       | r = 981       | c = where $2 \times 327 = 654$ and so $3 \times 327 = 981$ }} {{end-eqn}} {{OEIS|A039667}} It is interesting to note how the $4$ sums can be divided into two pairs in which each element of the sum is an anagram of its corresponding element in the other.


Primitive of x cubed over Root of x squared plus a squared cubed :$\displaystyle \int \frac {x^3 \ \mathrm d x} {\left({\sqrt {x^2 + a^2} }\right)^3} = \sqrt {x^2 + a^2} + \frac {a^2} {\sqrt {x^2 + a^2} } + C$


Primitive of Power of x by Power of Logarithm of x :$\displaystyle \int x^m \ln^n x \ \mathrm d x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \ \mathrm d x + C$ where $m \ne -1$.


Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 1 Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $\sequence {x_n}$,  $\sequence {y_n} $ be sequences in $R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Then: {{:Combination Theorem for Sequences/Normed Division Ring/Product Rule}}


Primitive of Hyperbolic Sine of a x over Power of x :$\displaystyle \int \frac {\sinh a x \ \mathrm d x} {x^n} = \frac {-\sinh a x} {\left({n - 1}\right) x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \ \mathrm d x} {x^{n - 1} } + C$


K-Connectivity Implies Lesser Connectivity If a graph $G$ is $k$-connected, then $G$ is $l$-connected for all $l \in \Z : 0 < l < k$.


Triangles with Proportional Sides are Similar Let two triangles have corresponding sides which are proportional. Then their corresponding angles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/5}}


Sum of Two Fifth Powers :$x^5 + y^5 = \paren {x + y} \paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$


Second Derivative of Convex Real Function is Non-Negative Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is convex on $\left({a \,.\,.\, b}\right)$ {{iff}} its second derivative $f'' \ge 0$ on $\left({a \,.\,.\, b}\right)$.


Partial Derivative wrt x of sin x y over cos (x + y) :$\dfrac \partial {\partial x} \dfrac {\sin x y} {\map \cos {x + y} } = \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$


Tartaglia's Formula Let $T$ be a tetrahedron with vertices $\mathbf d_1, \mathbf d_2, \mathbf d_3$ and $\mathbf d_4$. For all $i$ and $j$, let the distance between $\mathbf d_i$ and $\mathbf d_j$ be denoted $d_{ij}$. Then the volume $V_T$ of $T$ satisfies: :$V_T^2 = \dfrac {1} {288} \det \ \begin{vmatrix}   0 & 1 & 1 & 1 & 1\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$


Killing Form of Symplectic Lie Algebra Let $\mathbb K \in \left\{ {\C, \R}\right\}$. Let $n$ be a positive integer. Let $\mathfrak{sp}_{2 n} \left({\mathbb K}\right)$ be the Lie algebra of the symplectic group $\operatorname{Sp} \left({2 n, \mathbb K}\right)$. Then its Killing form is $B: \left({X, Y}\right) \mapsto \left({2 n + 2}\right) \operatorname{tr} \left({X Y}\right)$.


Group Action on Subgroup by Right Regular Representation is not Transitive Let $G$ be a group. Let $H$ be a proper subgroup of $G$. Let $*: H \times G \to G$ be the group action defined as: :$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$ where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$. Then $*$ is not transitive.


Intersection of Left Cosets of Subgroups is Left Coset of Intersection Let $G$ be a group. Let $H, K \le G$ be subgroups of $G$. Let $a, b \in G$. Let: :$a H \cap b K \ne \O$ where $a H$ denotes the left coset of $H$ by $a$. Then $a H \cap b K$ is a left coset of $H \cap K$.


Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0 The special case of Bessel's equation: :$(1): \quad x^2 y'' + x y' + \left({x^2 - \dfrac 1 4}\right) y = 0$ has the general solution: :$y = C_1 \dfrac {\sin x} {\sqrt x} + C_2 \dfrac {\cos x} {\sqrt x}$


Sequence of Powers of Number less than One/Rational Numbers Let $x \in \Q$. Let $\sequence {x_n}$ be the sequence in $\Q$ defined as $x_n = x^n$. Then: :$\size x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence.


Primitive of Cosine Integral Function :$\displaystyle \int \map \Ci x \rd x = x \map \Ci x + \sin x + C$


Sum from -m to m of Sine of n + alpha of theta over n + alpha For $0 < \theta < 2 \pi$: :$\displaystyle \sum_{n \mathop = -m}^m \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \cos {\alpha \theta} \dfrac {\sin \paren {m + \frac 1 2} \theta \rd \theta} {\sin \frac 1 2 \theta}$


Classical Probability is Probability Measure The classical probability model is a probability measure.


Subgroup Generated by Subgroup Let $G$ be a group. Let $H \le G$ be a subgroup of $G$. Then: :$H = \gen H$ where $\gen H$ denotes the subgroup generated by $H$.


Sufficient Condition for Vector Equals Inverse iff Zero Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms. Let $\mathbb F$ be infinite. Then: :$\forall \mathbf v, -\mathbf v \in \mathbf V: \mathbf v = - \mathbf v \iff \mathbf v = \mathbf 0$


Sum over k of m Choose k by k minus m over 2 :$\displaystyle \sum_{k \mathop = 0}^n \binom m k \left({k - \dfrac m 2}\right) = -\dfrac m 2 \binom {m - 1} n$ where $\dbinom m k$ etc. are binomial coefficients.


Equivalence of Definitions of Continuous Mapping between Topological Spaces/Everywhere Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$. {{TFAE|def = Everywhere Continuous Mapping Between Topological Spaces|view = everywhere continuous mapping between topological spaces}} === Definition by Pointwise Continuity === {{Definition:Continuous Mapping (Topology)/Everywhere/Pointwise}} === Definition by Open Sets === {{Definition:Continuous Mapping (Topology)/Everywhere/Open Sets}}


Primitive of x cubed by Sine of a x :$\displaystyle \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$


Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Proof 1 :$p \lor \left ({p \land q}\right) \dashv \vdash p$


Sum of Reciprocals in Base 10 with Zeroes Removed The infinite series :$\displaystyle \sum_{\map P n} \frac 1 n$ where $\map P n$ is the propositional function: :$\forall n \in \Z_{>0}: \map P n \iff$ the decimal representation of $n$ contains no instances of the digit $0$ converges to the approximate limit $23 \cdotp 10345 \ldots$


Diagonal Relation is Smallest Equivalence Relation The diagonal relation $\Delta_S$ on $S$ is the smallest equivalence in $S$, in the sense that: :$\forall \mathcal E \subseteq S \times S: \Delta_S \subseteq \mathcal E$ where $\mathcal E$ denotes a general equivalence relation.


Number of Matrix Equivalence Classes Let $K$ be a field. Let $\mathcal M_K \left({m, n}\right)$ be the $m \times n$ matrix space over $K$. Let $\mathbf A$ be an $m \times n$ matrix of rank $r$ over $K$. Then: :$\mathbf A \equiv \begin{cases} \left[{0_K}\right]_{m n} & : r = 0 \\ & \\ \begin{bmatrix}   \mathbf I_r & \mathbf 0 \\   \mathbf 0 & \mathbf 0  \end{bmatrix} & : 0 < r < \min \left\{{n, m}\right\} \\ & \\ \begin{bmatrix}   \mathbf I_r & \mathbf 0  \end{bmatrix} & : r = m < n \\ & \\ \begin{bmatrix}   \mathbf I_r \\   \mathbf 0  \end{bmatrix} & : r = n < m \\ & \\ \mathbf I_r & : r = m = n \end{cases}$ Thus there are exactly $\min \left\{{m, n}\right\} + 1$ equivalence classes for the relation of equivalence on $\mathcal M_K \left({m, n}\right)$, one of which contains only the zero matrix.


Complement of Top/Bounded Lattice Let $\left({S, \vee, \wedge, \preceq}\right)$ be a bounded lattice. Then the top $\top$ has a unique complement, namely $\bot$, bottom.


Summation of Product of Differences :$\displaystyle \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({u_j - u_k}\right) \left({v_j - v_k}\right) = n \sum_{j \mathop = 1}^n u_j v_j - \sum_{j \mathop = 1}^n u_j \sum_{j \mathop = 1}^n v_j$


Boundary of Subset of Discrete Space is Null Let $A \subseteq S$. Then: :$\partial A = \O$ where: :$\partial A$ is the boundary of $A$ in $T$.


Discrete Space is Complete Metric Space Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is a complete metric space.


Angle Between Vectors in Terms of Dot Product The angle between two non-zero  vectors in $\R^n$ can be calculated by: {{begin-eqn}} {{eqn | l = \theta       | r = \arccos \frac {\mathbf v \cdot \mathbf w} {\norm {\mathbf v} \norm {\mathbf w} } }} {{end-eqn}} where: :$\mathbf v \cdot \mathbf w$ represents the dot product of $\mathbf v$ and $\mathbf w$  :$\norm {\, \cdot \,}$ represents vector length. :$\arccos$ represents arccosine


Triangular Number cannot be Cube Let $T_n$ be the $n$th triangular number such that $n > 1$. Then $T_n$ cannot be a cube.


Triple Angle Formulas/Cosine/2 cos 3 theta + 1 :$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$


Matrix Space is Module Let $\struct {R, +, \circ}$ be a ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $\struct {R, +, \circ}$. Then the matrix space $\map {\MM_R} {m, n}$ of all $m \times n$ matrices over $R$ is a module.


Kernel of Group Homomorphism is not Empty Let $G$ and $H$ be groups whose identity elements are $e_G$ and $e_H$ respectively. Let $\phi: G \to H$ be a homomorphism from $G$ to $H$. Let $\map \ker \phi$ denote the kernel of $\phi$. Then: :$\map \ker \phi \ne \O$ where $\O$ denotes the empty set.


Linear Second Order ODE/2 y'' + 2 y' + 3 y = 0 The second order ODE: :$(1): \quad 2 y'' + 2 y' + 3 y = 0$ has the general solution: :$y = e^{-x/2} \paren {C_1 \cos \dfrac {\sqrt 5} 2 x + C_2 \sin \dfrac {\sqrt 5} 2 x}$


Topological Properties of Non-Archimedean Division Rings/Intersection of Closed Balls :$\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$


Set of Rational Numbers whose Numerator Divisible by p is Closed under Addition Let $p$ be a prime number. Let $A_p$ be the set of all rational numbers which, when expressed in canonical form has a numerator which is divisible by $p$. Then $A_p$ is closed under rational addition.


Metacategory Induces Morphisms-Only Metacategory Let $\mathbf C$ be a metacategory. Then the collection of morphisms $\mathbf C_1$ of $\mathbf C$ is a morphisms-only metacategory.


Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Sine x :$\displaystyle \int_0^{2 \pi} \frac {\d x} {\paren {a + b \sin x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$


Binomial Coefficient of Prime Plus One Modulo Prime Let $p$ be a prime number. Then: :$2 \le k \le p - 1 \implies \dbinom {p + 1} k \equiv 0 \pmod p$ where $\dbinom {p + 1} k$ denotes a binomial coefficient.


Normalizer of Subgroup of Symmetric Group that Fixes n Let $S_n$ denote the symmetric group on $n$ letters. Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that: :$\map \pi n = n$ The normalizer of $H$ is given by: :$\map {N_{S_n} } H = \map {N_{S_n} } {S_{n - 1} } = S_{n - 1}$


Primitive of Cube of Hyperbolic Cotangent of a x :$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$


Quotient of Ring of Polynomials in Ring Element on Integral Domain by that Polynomial is that Domain Let $\struct {D, +, \times}$ be an integral domain. Let $X \in R$ be transcencental over $D$. Let $D \sqbrk X$ be the ring of polynomials in $X$ over $D$. Let $D \sqbrk X / \ideal X$ denote the quotient ring of $D \sqbrk X$ by the ideal of $D$ generated by $X$. Then: :$D \sqbrk X / \ideal X \cong D$


Elements of Geometric Sequence from One where First Element is not Power of Number Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $k \in \Z_{> 1}$. Let $a_1$ not be a power of $k$. Then $a_m$ is not a power of $k$ except for: :$\forall m, k \in \set {1, 2, \ldots, n}: k \divides m$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/10}}


Primitive of Hyperbolic Cosine of a x over Power of x :$\ds \int \frac {\cosh a x \rd x} {x^n} = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C$


Hilbert Proof System Instance 2 Independence Results/RST4 is Derivable Rule of inference $RST \, 4$ is derivable from $RST \, 1, RST \, 2, RST \, 3$ and the axioms $(A1)$ through $(A4)$.


Fourier Series/Sawtooth Wave 600pxthumbrightSawtooth Wave and $6$th Approximation Let $\map S x$ be the sawtooth wave defined on the real numbers $\R$ as: :$\forall x \in \R: \map S x = \begin {cases} x & : x \in \openint {-l} l \\ \map S {x + 2 l} & : x < -l \\ \map S {x - 2 l} & : x > +l \end {cases}$ where $l$ is a given real constant. Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map S x       | o = \sim       | r = \frac {2 l} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \dfrac {n \pi x} l       | c =  }} {{eqn | r = \frac {2 l} \pi \paren {\sin \dfrac {\pi x} l - \frac 1 2 \sin \dfrac {2 \pi x} l + \frac 1 3 \sin \dfrac {3 \pi x} l + \dotsb}       | c =  }} {{end-eqn}}


Open Set in Partition Topology is Component Let $T = \struct {S, \tau}$ be a partition topological space. Then each of its open sets are components of $T$.


Basis for Finite Submodule of Function Space Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $A$ be a set. For each $a \in A$, let $f_a: A \to R$ be defined as: :$\forall x \in A: \map {f_a} x = \begin{cases} 1 & : x = a \\ 0 & : x \ne a \end{cases}$ Then $B = \set {f_a: a \in A}$ is a basis of the Finite Submodule of Function Space $R^{\paren A}$.


Order of Additive Group of Integers Modulo m Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. The order of $\struct {\Z_m, +_m}$ is $m$.


Continuum Property implies Well-Ordering Principle The Continuum Property of the positive real numbers $\R_{\ge 0}$ implies the Well-Ordering Principle of the natural numbers $\N$.


Strictly Monotone Mapping is Monotone A mapping that is strictly monotone is a monotone mapping.


Complex Number is Algebraic over Real Numbers Let $z \in \C$ be a complex number. Then $z$ is algebraic over $\R$.


Point is Isolated iff belongs to Set less Derivative Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Let $x \in S$. Then: :$x$ is an isolated point in $H$ {{iff}}: :$x \in H \setminus H'$ where :$H'$ denotes the derivative of $H$.


Sequence of Imaginary Reciprocals/Countability The set $S$ is countably infinite.


Positive Integer is Well-Defined "Positive" as applied to an integer is well-defined.


Orthogonal Trajectories/Rectangular Hyperbolas Consider the one-parameter family of curves of rectangular hyperbolas: :$(1): \quad x y = c$ Its family of orthogonal trajectories is given by the equation: :$x^2 - y^2 = c$ :600px


Dominated Strategy may be Optimal A dominated strategy of a game may be the optimal strategy for a player of that game.


Termial on Real Numbers is Extension of Integers The termial function as defined on the real numbers is an extension of its definition on the integers $\Z$.


Solution to Legendre's Differential Equation The solution of Legendre's differential equation: {{:Definition:Legendre's Differential Equation}} can be obtained by Power Series Solution Method. {{refactor|Include the actual solution here in the Theorem section and then proceed to derive each instance of that solution in the Proof section.  If necessary, split it up into bits.  At the moment it is too amorphous to be able to be followed easily.  My eyes are too sore to do a good job on this tonight so I won't.}}


Leigh.Samphier/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C$ be a dependent subset of $M$. Let $x \in S$. Let $X$ be an independent subset of $M$ such that: :$C \setminus \set x \subseteq X$. Then: :$x \notin X$


Eigenvalues of Normal Operator have Orthogonal Eigenspaces Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a normal operator. Let $\lambda, \mu$ be distinct eigenvalues of $A$. Then: :$\ker \left({A - \lambda}\right) \perp \ker \left({A - \mu}\right)$ where: : $\ker$ denotes kernel : $\perp$ denotes orthogonality.


Preimage of Set Difference under Mapping/Corollary 1 Let $f: S \to T$ be a mapping. Let $T_1 \subseteq T_2 \subseteq T$. Then: :$\complement_{f^{-1} \left[{T_2}\right]} \left({f^{-1} \left[{T_1}\right]}\right) = f^{-1} \left[{\complement_{T_2} \left({T_1}\right)}\right]$ where: : $\complement$ (in this context) denotes relative complement : $f^{-1} \left[{T_1}\right]$ denotes preimage.


Square of Repunit times Sum of Digits The following pattern emerges: {{begin-eqn}} {{eqn | l = 121 \times \paren {1 + 2 + 1}       | r = 22^2 }} {{eqn | l = 12 \, 321 \times \paren {1 + 2 + 3 + 2 + 1}       | r = 333^2 }} {{eqn | l = 1 \, 234 \, 321 \times \paren {1 + 2 + 3 + 4 + 3 + 2 + 1}       | r = 4444^2 }} {{end-eqn}} and so on, up until $999 \, 999 \, 999^2$ after which the pattern breaks down.


Countable Open Ordinal Space is Metrizable Let $\Omega$ denote the first uncountable ordinal. Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$. Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$. Then $\hointr 0 \Gamma$ is a metrizable space.


Composite Number has Prime Factor not Greater Than its Square Root Let $n \in \N$ and $n = p_1 \times p_2 \times \cdots \times p_j$, $j \ge 2$, where $p_1, \ldots, p_j \in \Bbb P$ are prime factors of $n$. Then $\exists p_i \in \Bbb P$ such that $p_i \le \sqrt n$. That is, if $n \in \N$ is composite, then $n$ has a prime factor $p \le \sqrt n$.


Satisfiable Set minus Formula is Satisfiable Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$. Let $\phi \in \mathcal F$. Then $\mathcal F \setminus \left\{{\phi}\right\}$ is also $\mathscr M$-satisfiable.


Primitive of Inverse Hyperbolic Cosecant of x over a :$\displaystyle \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x = \begin{cases} x \operatorname{csch}^{-1} \dfrac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\ x \operatorname{csch}^{-1} \dfrac x a - a \sinh^{-1} \dfrac x a + C & : x < 0 \end{cases}$


Fort Space is T0 Let $T = \left({S, \tau_p}\right)$ be a Fort space on an infinite set $S$. Then $T$ is a $T_0$ (Kolmogorov) space.


Power Series Expansion for Cosine Integral Function plus Logarithm :$\displaystyle \map \Ci x = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^{2 n} } {\paren {2 n} \times \paren {2 n}!}$


Pullback Functor is Functor Let $\mathbf C$ be a metacategory having all pullbacks. Let $f: C \to D$ be a morphism of $\mathbf C$. Let $\mathbf C \mathbin / C$ and $\mathbf C \mathbin / D$ be the slice categories over $C$ and $D$, respectively. Let $f^* : \mathbf C \mathbin / D \to \mathbf C \mathbin / C$ be the pullback functor defined by $f$. Then $f^*$ is a functor.


Primitive of Hyperbolic Cosine of a x over x squared :$\displaystyle \int \frac {\cosh a x \ \mathrm d x} {x^2} = -\frac {\cosh a x} x + a \int \frac {\sinh a x \ \mathrm d x} x$


Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space. Let $Y \subseteq X$ be a vector subspace. Let $\norm {\, \cdot \,}_Y$ be the induced norm on $Y$. Then $\struct {Y, \norm {\, \cdot \,}_Y}$ is a normed vector space.


Image of Union under Mapping/General Result Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let $\powerset S$ be the power set of $S$. Let $\mathbb S \subseteq \powerset S$. Then: :$\displaystyle f \sqbrk {\bigcup \mathbb S} = \bigcup_{X \mathop \in \mathbb S} f \sqbrk X$


Stabilizer of Subgroup Action is Identity Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as: :$\forall h \in H, g \in G: h * g := h \circ g$ The stabilizer of $x \in G$ is $\set e$: :$\Stab x = \set e$


Integral with respect to Dirac Measure Let $\left({X, \Sigma}\right)$ be a measurable space. Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$. Let $f \in \mathcal M _{\overline \R}, f: X \to \overline \R$ be a measurable function. Then: :$\displaystyle \int f \, \mathrm d \delta_x = f \left({x}\right)$ where the integral sign denotes the $\delta_x$-integral.


Equivalence of Definitions of Generated Subgroup {{TFAE|def = Generated Subgroup}} Let $G$ be a group. Let $S \subset G$ be a subset.


Graph Components are Equivalence Classes The components of a graph are equivalence classes under the relation '''is connected to''' on the set of vertices.


Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars It is not possible to extend the complex numbers to an algebra of $3$ dimensions with real scalars.


Smallest Titanic Palindromic Prime The smallest titanic prime that is also palindromic is: :$10^{1000} + 81 \, 918 \times 10^{498} + 1$ which can be written as: :$1 \underbrace {000 \ldots 000}_{497} 81918 \underbrace {000 \ldots 000}_{497} 1$


Parity Group is Group The parity group is in fact a group.


At Most Two Horizontal Asymptotes The graph of a real function has at most two horizontal asymptotes.


P-adic Valuation Extends to P-adic Numbers Let $p$ be a prime number. Let $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on the set of  rational numbers. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ be defined by: :$\forall x \in \Q_p : \map {\nu_p} x = \begin {cases} -\log_p \norm {x}_p : x \ne 0 \\ +\infty : x = 0 \end{cases}$ Then $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ is a valuation that extends $\nu_p^\Q$ from $\Q$ to $\Q_p$.


Odd Numbers not Sum of Prime and Power The sequence of odd numbers which cannot be expressed as the sum of a perfect power and a prime number begins: :$1, 5, 1549, 1 \, 771 \, 561, \ldots$ {{OEIS|A119747}} It is not known if there are any more terms.


Modulo Operation/Examples/-2 mod 3 :$-2 \bmod 3 = 1$


Geometrical Interpretation of Complex Modulus Let $z \in \C$ be a complex number expressed in the complex plane. Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin.


Linear First Order ODE/y' + y cot x = 2 x cosec x The linear first order ODE: :$(1): \quad y' + y \cot x = 2 x \csc x$ has the general solution: :$y = x^2 \csc x + C \csc x$


Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity Let $G$ be a group whose identity is $e_G$. Let $H$ be a group whose identity is $e_H$. Let $\phi: G \to H$ be a (group) homomorphism. Let $x^n \in \map \ker \phi$ for some integer $n$. Then: :$\paren {\map \phi x}^n = e_H$


Non-Zero Natural Numbers under Multiplication form Commutative Monoid Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \set 0$. The structure $\struct{\N_{>0}, \times}$ forms a commutative monoid.


Rational Number Space is Topological Space Let $\struct {\Q, \tau_d}$ be the rational number space formed by the rational numbers $\Q$ under the usual (Euclidean) topology $\tau_d$. Then $\tau_d$ forms a topology.


Inverse Hyperbolic Cosine of Imaginary Number :$\cosh^{-1} x = \pm \, i \cos^{-1} x$


Finite Group whose Subsets form Nest is Cyclic P-Group Let $G$ be a group. Let $G$ be such that its subgroups form a nest. Then $G$ is a cyclic $p$-group.


Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Sine :$\displaystyle \int \frac {\d x} {\sin^m a x \cos^n a x} = \frac {-1} {a \paren {n - 1} \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {m - 1} \int \frac {\d x} {\sin^{m - 2} a x \cos^n a x} + C$


Floor Function/Examples/Floor of Root 5 :$\floor {\sqrt 5} = 2$


Function is Odd Iff Inverse is Odd Let $f$ be an odd real function with an inverse $f^{-1}$. Then $f^{-1}$ is also odd.


Mapping at Element is Supremum of Compact Elements implies Mapping is Increasing Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ be a lattice. Let $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be a complete lattice. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \preceq_1 x \land w}\right.$ is compact$\left.{}\right\}$ Then $f$ is increasing.


Quotient Group of Ideal is Coset Space Let $\struct {R, +, \circ}$ be a ring. Let $J$ be an ideal of $R$. Let $\struct {R / J, +}$ be the quotient group of $\struct {R, +}$ by $\struct {J, +}$. Then each element of $\struct {R / J, +}$ is a coset of $J$ in $R$, that is, is of the form $x + J = \set {x + j: j \in J}$ for some $x \in R$. The rule of addition of these cosets is: $\paren {x + J} + \paren {y + J} = \paren {x + y} + J$. The identity of $\struct {R / J, +}$ is $J$ and for each $x \in R$, the inverse of $x + J$ is $\paren {-x} + J$.


Rational Number Space is Second-Countable Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is second-countable.


Existence of Non-Square Residue Let $m \in \Z$ be an integer such that $m > 2$. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m - 1}\right]\!\right]_m}\right\}$ Then there exists at least one residue in $\Z_m$ which is not the product modulo $m$ of a residue with itself: :$\exists p \in \Z_M: \forall x \in \Z_m: x \cdot_m x \ne p$


Positive Multiple of Metric is Metric Let $M = \left({A, d}\right)$ be a metric space. Let $k \in \R_{>0}$ be a (strictly) positive real number. Let $d_k: A \times A \to \R$ be the function defined as: :$\forall \left({x, y}\right) \in A: d_k \left({x, y}\right) = k \cdot d \left({x, y}\right)$ Then $M_k = \left({A, d_k}\right)$ is a metric space.


Construction of Smith Number from Prime Repunit Let $R_n$ be a repunit which is prime where $n \ge 3$. Then $3304 \times R_n$ is a Smith number. $3304$ is not the only number this works for, but it is the smallest.


Zero Element of Multiplication on Numbers On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ the zero element of multiplication is zero ($0$).


Odd Integers whose Smaller Odd Coprimes are Prime Let $n \in \Z_{>0}$ be an odd positive integer such that all smaller odd integers greater than $1$ which are coprime to it are prime. The complete list of such $n$ is as follows: :$1, 3, 5, 7, 9, 15, 21, 45, 105$ {{OEIS|A327823}}


Reparameterization of Contour is Contour Let $\closedint a b$ and $\closedint c d$ be closed real intervals. Let $\gamma: \closedint a b \to \C$ be a contour in $\C$. That is, there exists a subdivision $a_0, a_1 , \ldots, a_n$ of $\closedint a b$ such that $\gamma \restriction_{I_i}$ is a smooth path for all $i \in \set {1, \ldots, n}$, where $I_i = \closedint {a_{i - 1} } {a_i}$. Here, $\gamma \restriction_{I_i}$ denotes the restriction of $\gamma$ to $I_i$. Let $\phi: \closedint c d \to \closedint a b$  be a differentiable real function with $\map \phi c = a$, and $\map \phi d = b$. Suppose that $\phi$ is either bijective or strictly increasing. Define $\sigma: \closedint c d \to \C$ as a composite function by $\sigma = \gamma \circ \phi$. Then $\sigma$ is a contour. For all $i \in \set {1, \ldots, n}$, let $J_i := \closedint {\map {\phi^{-1} } {a_{i - 1} } } {\map {\phi^{-1} } {a_i} }$. Then: :$\map {\sigma' \restriction_{J_i} } t = \map {\gamma' \restriction_{I_i} } {\map \phi t} \map {\phi'} t$


Power Series Expansion for Fresnel Sine Integral Function :$\displaystyle \map {\operatorname S} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}$


Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary Let $R$ be a commutative ring. Let $\struct {S, +, *}$ be a ring with unity. Let $f: R \to S$ be a ring homomorphism. Let $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$. Then $\struct {S_R, *}$ is a unitary algebra.


Sine in terms of Tangent {{begin-eqn}} {{eqn | l = \sin x       | r = +\frac {\tan x} {\sqrt {1 + \tan^2 x} }       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \sin x       | r = -\frac {\tan x} {\sqrt {1 + \tan^2 x} }       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}


Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3 Let $S_3$ denote the Symmetric Group on 3 Letters. Let $D_3$ denote the dihedral group $D_3$. Then $S_3$ is isomorphic to $D_3$.


Squares of Diagonals of Parallelogram Let $ABCD$ be a parallelogram. :400px Then: :$AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$


Quaternion Group not Dihedral Group Let $Q$ be the quaternion group. Then $Q$ is not isomorphic to the dihedral group $D_4$.


Mappings Partially Ordered by Extension Let $S$ and $T$ be sets. Let $F$ be the set of all mappings from $S$ to $T$. Let $\mathcal R \subseteq F \times F$ be the relation defined as: :$\left({f, g}\right) \in \mathcal R \iff \operatorname{Dom} \left({f}\right) \subseteq \operatorname{Dom} \left({g}\right) \land \forall x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = g \left({x}\right)$ That is, $f \mathrel{\mathcal R} g$ {{iff}} $g$ is an extension of $f$. Then $\mathcal R$ is an ordering on $F$.


Positive Real Axis forms Subgroup of Complex Numbers under Multiplication Let $S$ be the subset of the set of complex numbers $\C$ defined as: :$S = \set {z \in \C: z = x + 0 i, x > 0}$ That is, let $S$ be the positive real axis of the complex plane. Then the algebraic structure $\struct {S, \times}$ is a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.


P-adic Valuation of Rational Number is Well Defined The $p$-adic valuation: : $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is well defined.


Limit of Sets Exists iff Limit Inferior contains Limit Superior Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets. Then $\Bbb S$ converges to a limit {{iff}}: :$\ds \limsup_{n \mathop \to \infty} E_n \subseteq \liminf_{n \mathop \to \infty}E_n$


Ceiling Function/Examples/Ceiling of Root 2 :$\ceiling {\sqrt 2} = 2$


Inverse Image Mapping Induced by Projection Let $\family {S_i}_{i \mathop \in I}$ be a family of sets. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. For each $j \in I$, let $\pr_j: S \to S_j$ denote the $j$-th projection. For each $j \in I$ let $\pr_j^\gets: \powerset {S_i} \to \powerset S$ denote the inverse image mapping induced by $\pr_j$ Then for all $j \in I$, $\pr_j^\gets$ is the mapping defined by: :$\forall T \subseteq S_i: \map {\pr_j^\gets} T = \displaystyle \prod_{i \mathop \in I} T_i$ where: :$T_i = \begin {cases} T & i = j \\ S_i & i \ne j \end {cases}$


Condition for Commensurability of Roots of Quadratic Equation Consider the quadratic equation: :$(1): \quad a x - x^2 = \dfrac {b^2} 4$ Then $x$ and $a - x$ are commensurable {{iff}} $\sqrt{a^2 - b^2}$ and $a$ are commensurable. {{:Euclid:Proposition/X/17}}


Measurable Functions Determine Measurable Sets Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g: X \to \overline \R$ be $\Sigma$-measurable functions. Then the following sets are measurable: :$\set {f < g}$ :$\set {f \le g}$ :$\set {f = g}$ :$\set {f \ne g}$ where, for example, $\set {f < g}$ is short for $\set {x \in X: \map f x < \map g x}$.


C6 is not Isomorphic to S3 Let $C_6$ denote the cyclic group of order $6$. Let $S_3$ denote the symmetric group on $3$ letters. Then $C_6$ and $S_3$ are not isomorphic.


Primitive of Root of x squared minus a squared over x squared :$\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$


Set of Even Integers is Equivalent to Set of Integers Let $\Z$ denote the set of integers. Let $2 \Z$ denote the set of even integers. Then: :$2 \Z \sim \Z$ where $\sim$ denotes set equivalence.


Primitive of Arctangent of x over a over x squared :$\displaystyle \int \frac {\arctan \frac x a \ \mathrm d x} {x^2} = \frac {-\arctan \frac x a} x - \frac 1 {2 a} \ln \left({\frac {x^2 + a^2} {x^2} }\right) + C$


Row in Pascal's Triangle forms Palindromic Sequence Each of the rows of Pascal's triangle forms a palindromic sequence.


Negative Binomial Distribution as Generalized Geometric Distribution/First Form The first form of the negative binomial distribution is a generalization of the geometric distribution: Let $\left \langle{X_i}\right \rangle$ be a Bernoulli process with parameter $p$. Let $\mathcal E$ be the experiment which consists of: : Perform the Bernoulli trial $X_i$ until $n$ failures occur, and then stop. Let $k$ be the number of successes before before $n$ failures have been encountered. Let $\mathcal E'$ be the experiment which consists of: : Perform the Bernoulli trial $X_i$ until '''one''' failure occurs, and then stop. Then $k$ is modelled by the experiment: : Perform experiment $\mathcal E'$ until $n$ failures occur, and then stop.


Solution of Linear 2nd Order ODE Tangent to X-Axis Let $\map {y_p} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ on a closed interval $\closedint a b$. Let there exist $\xi \in \closedint a b$ such that the curve in the cartesian plane described by $y = \map {y_p} x$ is tangent to the $x$-axis at $\xi$. Then $\map {y_p} x$ is the zero constant function: :$\forall x \in \closedint a b: \map {y_p} x = 0$


Differentiable Function with Bounded Derivative is of Bounded Variation Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a continuous function. Let $f$ be differentiable on $\openint a b$, with bounded derivative. Then $f$ is of bounded variation.


Set Intersection is not Cancellable Set intersection is not a cancellable operation. That is, for a given $A, B, C \subseteq S$ for some $S$, it is not always the case that: :$A \cap B = A \cap C \implies B = C$


Sum of Series of Product of Power and Cosine Let $r \in \R$. Let $x \in \R$ such that $x \ne 2 m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^n r^k \map \cos {k x}       | r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots + r^n \cos n x       | c =  }} {{eqn | r = \dfrac {r^{n + 2} \cos n x - r^{n + 1} \map \cos {n + 1} x - r \cos x + 1} {1 - 2 r \cos x + r^2}       | c =  }} {{end-eqn}}


Transitive Relation whose Symmetric Closure is not Transitive Let $S = \set {p, q}$, where $p$ and $q$ are distinct elements. Let $\RR = \set {\tuple {p, q} }$. Then $\RR$ is transitive but its symmetric closure is not.


Equivalence of Definitions of Minimal Polynomial {{TFAE|def = Minimal Polynomial}} Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$.


Image of Successor Mapping forms Peano Structure Let $\left({P, s, 0}\right)$ be a Peano structure. Let $P'$ be the set $s \left[{P}\right]$, that is: :$P' = \left\{{s \left({n}\right): n \in P}\right\}$ Let $s'$ be the restriction of $s$ to $P'$. Then $\left({P', s', s \left({0}\right)}\right)$ is also a Peano structure.


Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space Let $\R^n$ be an n-dimensional real vector space. Let $\norm {\, \cdot \,}_2$ be the Euclidean norm. Then $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a normed vector space.


Laplace's Expansion Theorem Let $D$ be the determinant of order $n$. Let $r_1, r_2, \ldots, r_k$ be integers such that: :$1 \le k < n$ :$1 \le r_1 < r_2 < \cdots < r_k \le n$ Let $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be an order-$k$ minor of $D$. Let $\map {\tilde D} {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be the cofactor of $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$. Then: :$\displaystyle D = \sum_{1 \mathop \le u_1 \mathop < \cdots \mathop < u_k \mathop \le n} \map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k} \, \map {\tilde D} {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ A similar result applies for columns.


Semantic Consequence as Tautological Conditional Let $\mathcal F$ be a finite set of WFFs of propositional logic. Let $\mathbf A$ be another WFF. Then the following are equivalent: {{begin-eqn}} {{eqn|l = \mathcal F      |o = \models_{\mathrm{BI} }      |r = \mathbf A }} {{eqn|o = \models_{\mathrm{BI} }       |r = \bigwedge \mathcal F \implies \mathbf A }} {{end-eqn}} that is, $\mathbf A$ is a semantic consequence of $\mathcal F$ iff $\displaystyle \bigwedge \mathcal F \implies \mathbf A$ is a tautology. Here, $\displaystyle \bigwedge \mathcal F$ is the conjunction of $\mathcal F$.


Half-Open Real Interval is Closed in some Open Intervals Let $\R$ be the real number line considered as an Euclidean space. Let $\left[{a \,.\,.\, b}\right) \subset \R$ be a half-open interval of $\R$. Let $c < a$. Then $\left[{a \,.\,.\, b}\right)$ is a closed set of $\left({c \,.\,.\, b}\right)$. Similarly, let $d > b$. Then the half-open interval $\left({a \,.\,.\, b}\right]$ is a closed set of $\left({a \,.\,.\, d}\right)$.


Unity of Subring is not necessarily Unity of Ring Let $\struct {S, +, \circ}$ be a ring with unity whose unity is $1_S$. Let $\struct {T, + \circ}$ be a subring of $\struct {S, + \circ}$ whose unity is $1_T$. Then it is not necessarily the case that $1_T = 1_S$.


Image of Real Square Function The image of the real square function is the entire set of positive real numbers $\R_{\ge 0}$.


Stirling Number of the Second Kind of Number with Greater Let $\displaystyle {n \brace k}$ denote a Stirling number of the second kind. Then: :$\displaystyle {n \brace k} = 0$


Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation Let $S$ be a set. Let $\RR \subseteq S \times S$ be a relation in $S$. Then: :$\RR$ is reflexive, symmetric and antisymmetric {{iff}}: :$\RR$ is the diagonal relation $\Delta_S$.


Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series Let $f, g: \N \to \C$ be arithmetic functions with Dirichlet convolution $h = f * g$. Let $F, G, H$ be their Dirichlet series. Let $\sigma_f, \sigma_g, \sigma_h$ be their abscissae of absolute convergence. Then: :$\sigma_h \le \max \set {\sigma_f, \sigma_g}$


Number of Components after Removal of Bridge Let $G = \left({V, E}\right)$ be a graph. Let $e \in E$ be a bridge. Let $m$ be the number of components of $G$. Then when $e$ is removed from $G$, the number of components in the remaining graph is $m + 1$.


Real Part of Complex Product Let $z_1$ and $z_2$ be complex numbers. Then: :$\map \Re {z_1 z_2} = \map \Re {z_1} \, \map \Re {z_2} - \map \Im {z_1} \, \map \Im {z_2}$


Dirichlet Convolution is Commutative : $f * g = g * f$


Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined Let the $\times_\beta$ operator ("multiplication") on the $\beta$-equivalence classes be defined as: :$\eqclass a \beta \times_\beta \eqclass b \beta := \eqclass {a \times b} \beta$ Then such an operation is well-defined.


Set is Equivalent to Proper Subset of Power Set Every set is equivalent to a proper subset of its power set: : $\forall S: \exists T \subset \mathcal P \left({S}\right): S \sim T$


Projection is Surjection/General Version For all non-empty sets $S_1, S_2, \ldots, S_j, \ldots, S_n$, the $j$th projection $\operatorname{pr}_j$ on $\displaystyle \prod_{i \mathop = 1}^n S_i$ is a surjection.


Bijection between R x (S x T) and (R x S) x T Let $R$, $S$ and $T$ be sets. Let $S \times T$ be the Cartesian product of $S$ and $T$. Then there exists a bijection from $R \times \paren {S \times T}$ to $\paren {R \times S} \times T$. Hence: :$\card {R \times \paren {S \times T} } = \card {\paren {R \times S} \times T}$


Necessary and Sufficient Condition for First Order System to be Field for Functional Let $\mathbf y$ be an N-dimensional vector. Let $J$ be a (real) functional such that: :$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$ Let the corresponding momenta and Hamiltonian be: {{begin-eqn}} {{eqn | l = \map {\mathbf p} {x, \mathbf y, \mathbf y'}       | r = \dfrac {\partial \map F {x, \mathbf y, \mathbf y'} } {\partial \mathbf y'} }} {{eqn | l = \map H {x, \mathbf y, \mathbf y'}       | r = -\map F {x, \mathbf y, \mathbf y'} + \mathbf p \mathbf y' }} {{end-eqn}} Let the following be a family of boundary conditions: :$(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$ Then a family of boundary conditions is a field for the functional $J$ {{iff}} $\forall x \in \closedint a b$ the following self-adjointness and consistency relations hold: {{begin-eqn}} {{eqn | l = \dfrac {\partial \map {p_i} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial y_k}       | r = \dfrac {\partial \map {p_k} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial y_i} }} {{eqn | l = \dfrac {\partial \map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x}       | r = -\dfrac {\partial \map H {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial \mathbf y} }} {{end-eqn}} {{explain|Why "consistency"?}} {{explain|Meaning of square brackets, above and throughout}}


Fréchet Space (Functional Analysis) is Complete Metric Space Let $\struct {\R^\omega, d}$ be the '''Fréchet space on $\R^\omega$'''. Then $\struct {\R^\omega, d}$ is a complete metric space.


Sum of Sequence of k x k! Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n j \times j!       | r = 1 \times 1! + 2 \times 2! + 3 \times 3! + \dotsb + n \times n!       | c =  }} {{eqn | r = \paren {n + 1}! - 1       | c =  }} {{end-eqn}}


Elements of Geometric Sequence from One which are Powers of Number Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Then: :$\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a power of $k$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/8}}


Best Rational Approximations to Root 2 generate Pythagorean Triples Consider the Sequence of Best Rational Approximations to Square Root of 2: :$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$ Every other term of $\sequence S$ can be expressed as: :$\dfrac {2 a + 1} b$ such that: :$a^2 + \left({a + 1}\right)^2 = b^2$ :$b$ is odd.


Line Parallel to Base of Trapezoid which Bisects Leg also Bisects Other Leg Let $ABCD$ be a trapezoid whose bases are $AB$ and $CD$. Let $EF$ be a straight line parallel to $AB$ and $CD$ which intersects its legs $AD$ and $BC$. Let $EF$ bisect $AD$. Then $EF$ also bisects $BC$.


Bijection iff exists Mapping which is Left and Right Inverse Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then $f$ is a bijection {{iff}}: :there exists a mapping $g: T \to S$ such that: ::$g \circ f = I_S$ ::$f \circ g = I_T$ :where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively.


Numbers Divisible by Sum and Product of Digits The sequence of positive integers which are divisible by both the sum and product of its digits begins: :$1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 111, 112, 132, 135, \ldots$ {{OEIS|A038186}}


Excluded Point Space is not Locally Arc-Connected Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is not locally arc-connected.


Characterization of Stationary Gaussian Process Let $S$ be a Gaussian stochastic process giving rise to a time series $T$. Let the the mean of $S$ be fixed. Let the autocovariance matrix of $S$ also be fixed. Then $S$ is stationary.


Unique Constant in Category of Monoids Let $\mathbf{Mon}$ be the category of monoids. Then every object $M$ of $\mathbf{Mon}$ has precisely one constant.


Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y') Let $F$ and its partial derivatives $F_y, F_{y'}$ be real functions, defined on the closed interval $I = \closedint a b$. Let $F, F_y, F_{y'} $ be continuous at every point $\tuple {x, y}$ for all finite $y'$. Suppose there exists a constant $k > 0$ such that: :$\map {F_y} {x, y, y'} > k$ Suppose there exist real functions $\alpha = \map \alpha {x, y} \ge 0$, $\beta = \map \beta {x, y}\ge 0$ bounded in every bounded region of the plane such that: :$\size {\map F {x, y, y'} } \le \alpha y'^2 + \beta$ Then one and only one integral curve of equation $y'' = \map F {x, y, y'}$ passes through any two points $\tuple {a, A}$ and $\tuple {b, B}$ such that $a \ne b$.


Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers Let $F_n$ denote the $n$th Fibonacci number. Then: :$F_{3 n} = {F_{n + 1} }^3 + {F_n}^3 - {F_{n - 1} }^3$


Cycle Matroid is Matroid Let $G = \struct{V,E}$ be a graph. Let $\struct{E, \mathscr I}$ be the cycle matroid of $G$. Then $\struct{E, \mathscr I}$ is a matroid.


Smallest Magic Constant of Order 3 Multiplicative Magic Square The magic constant of the smallest possible multiplicative magic square with the smallest magic constant is as follows. {{:Multiplicative Magic Square/Examples/Order 3/Smallest}} </onlyinclude>


Derivative of Cotangent of Function :$\map {\dfrac \d {\d x} } {\cot u} = -\csc^2 u \dfrac {\d u} {\d x}$


Successor Mapping is Inflationary Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction. Let $s: \omega \to \omega$ denote the successor mapping on $\omega$. Then $s$ is an inflationary mapping.


Measure Space Sigma-Finite iff Cover by Sets of Finite Measure Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\left({X, \Sigma, \mu}\right)$ is $\sigma$-finite iff there exists a sequence $\left({E_n}\right)_{n \in \N}$ in $\Sigma$ such that: :$(1):\quad \displaystyle \bigcup_{n \mathop \in \N} E_n = X$ :$(2):\quad \forall n \in \N: \mu \left({E_n}\right) < +\infty$


Taylor Series reaches closest Singularity Let the singularities of a function be the points at which the function is not analytic. Let $F$ be a complex function. Let $F$ be analytic everywhere except at a finite number of singularities. Let $x_0$ be a point in $\R$ where $F$ is analytic. Let $R \in \R_{>0}$ be the distance from $x_0$ to the closest singularity of $F$. Let $f = F {\restriction_\R}$ be a real function. Then: :the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \in \R$ satisfying $\size {x - x_0} < R$


Odd Power Function is Strictly Increasing/Real Numbers Let $n \in \Z_{> 0}$ be an odd positive integer. Let $f_n: \R \to \R$ be the real function defined as: :$\map {f_n} x = x^n$ Then $f_n$ is strictly increasing.


Infinite Intersection of Open Sets of Metric Space may not be Open Let $M = \struct {A, d}$ be a metric space. Let $U_1, U_2, U_3, \ldots$ be an infinite set of open sets of $M$. Then it is not necessarily the case that $\displaystyle \bigcap_{n \mathop \in \N} U_n$ is itself an open set of $M$.


Unbounded Monotone Sequence Diverges to Infinity/Increasing Let $\sequence {x_n}$ be increasing and unbounded above. Then $x_n \to +\infty$ as $n \to \infty$.


Power Series Expansion for Logarithm of 1 + x over 1 + x {{begin-eqn}} {{eqn | l = \frac {\map \ln {1 + x} } {1 + x}       | r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} H_n x^n       | c =  }} {{eqn | r = x - H_2 x^2 + H_3 x^3 - H_4 x^4 + \cdots       | c =  }} {{end-eqn}} where $H_n$ denotes the $n$th harmonic number: :$H_n = \ds \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 \cdots + \dfrac 1 r$ valid for all $x \in \R$ such that $\size x < 1$.


Regular Heptagon is Smallest with no Compass and Straightedge Construction The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction.


Infimum Precedes Coarser Infimum Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X, Y$ be subsets of $S$ such that :$Y$ is coarser than $X$. Then $\inf X \preceq \inf Y$ where $\inf X$ denotes the infimum of $X$.


Complement of Limit Superior is Limit Inferior of Complements Let $\left\{{E_n : n \in \N}\right\}$ be a sequence of sets. Then: :$\displaystyle \complement \left({\limsup_{n \to \infty} \ E_n}\right) = \liminf_{n \to \infty} \ \complement \left({E_n}\right)$ where $\limsup$ and $\liminf$ denote the limit superior and limit inferior, respectively.


Area inside Cardioid Consider the cardioid $C$ embedded in a polar plane given by its polar equation: :$r = 2 a \paren {1 + \cos \theta}$ The area inside $C$ is $6 \pi a^2$.


Image of Intersection under Injection/General Result Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Then: :$\displaystyle \forall \mathbb S \subseteq \mathcal P \left({S}\right): f \left[{\bigcap \mathbb S}\right] = \bigcap_{X \mathop \in \mathbb S} f \left[{X}\right]$  {{iff}} $f$ is an injection.


Point in Metric Space has Countable Neighborhood Basis Let $M = \left({A, d}\right)$ be a metric space. Let $a \in A$. Then there exists a basis for the neighborhood system of $a$ which is countable.


Linear Second Order ODE/x^2 y'' + 3 x y' + 10 y = 0 The second order ODE: :$(1): \quad x^2 y'' + 3 x y' + 10 y = 0$ has the general solution: :$y = \dfrac 1 x \paren {C_1 \, \map \cos {\ln x^3} + C_2 \, \map \sin {\ln x^3} }$


Preimage of Set Difference under Relation Let $\mathcal R \subseteq S \times T$ be a relation. Let $C$ and $D$ be subsets of $T$. Then: :$\mathcal R^{-1} \sqbrk C \setminus \mathcal R^{-1} \sqbrk D \subseteq \mathcal R^{-1} \sqbrk {C \setminus D}$ where: :$\setminus$ denotes set difference :$\mathcal R^{-1} \sqbrk C$ denotes the preimage of $C$ under $\mathcal R$.


Evolute of Parabola The evolute of the parabola $y = x^2$ is the curve: :$27 X^2 = 16 \paren {Y - \dfrac 1 2}^3$


Set of Isolated Points of Metric Space is Disjoint from Limit Points Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H'$ be the set of limit points of $H$. Let $H^i$ be the set of isolated points of $H$. Then: :$H' \cap H^i = \varnothing$


Product of Consecutive Triangular Numbers :$2 T_n T_{n - 1} = T_{n^2 - 1}$ where $T_n$ denotes the $n$th triangular number.


Eccentricity of Orbit indicates its Total Energy Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to. Then the total energy of the system determines the eccentricity of the orbit of $p$ around $S$.


Existence of Greatest Common Divisor Let $a, b \in \Z$ be integers such that $a \ne 0$ or $b \ne 0$. Then the greatest common divisor of $a$ and $b$ exists.


Composite of Injection on Surjection is not necessarily Either Let $f$ be an injection. Let $g$ be a surjection. Let $f \circ g$ denote the composition of $f$ with $g$. Then it is not necessarily the case that $f \circ g$ is either a surjection or an injection.


Directed Hamilton Cycle Problem is NP-complete Both versions of the Directed Hamilton Cycle problem are NP-complete.


Primitive of x cubed over a x + b cubed :$\displaystyle \int \frac {x^3 \rd x} {\paren {a x + b}^3} = \frac x {a^3} - \frac {3 b^2} {a^4 \paren {a x + b} } + \frac {b^3} {2 a^4 \paren {a x + b}^2} - \frac {3 b} {a^4} \ln \size {a x + b} + C$


Smallest Prime Number not Difference between Power of 2 and Power of 3 $41$ is the smallest prime number which is not the difference between a power of $2$ and a power of $3$.


Ramanujan Sum is Multiplicative Let $q \in \N_{>0}$, $n \in \N$. Let $\map {c_q} n$ be the Ramanujan sum. Then $\map {c_q} n$ is multiplicative in $q$.


Sum over k of n Choose k by x to the k by kth Harmonic Number/x = -1 While for $x \in \R_{> 0}$ be a real number: :$\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \left({x + 1}\right)^n \left({H_n - \ln \left({1 + \frac 1 x}\right)}\right) + \epsilon$ when $x = -1$ we have: :$\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \dfrac {-1} n$ where: : $\dbinom n k$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number.


Numbers whose Cube equals Sum of Sequence of that many Squares The integers $m$ in the following sequence all have the property that $m^3$ is equal to the sum of $m$ consecutive squares: :$m^3 = \displaystyle \sum_{k \mathop = 1}^m \left({n + k}\right)^2$ for some $n \in \Z_{\ge 0}$: :$0, 1, 47, 2161, 99 \, 359, 4 \, 568 \, 353, \ldots$ {{OEIS|A189173}}


Derivative at Point of Inflection Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$. Let $f$ have a point of inflection at $\xi \in \left({a \,.\,.\, b}\right)$. Then: :$f'' \left({\xi}\right) = 0$ where $f'' \left({\xi}\right)$ denotes the second derivative of $f$ at $\xi$.


Field Homomorphism Preserves Product Inverses Let $\phi: \struct {F_1, +_1, \times_1} \to \struct {F_2, +_2, \times_2}$ be a field homomorphism. Then: :$\forall x \in F_1^*: \map \phi {x^{-1} } = \map \phi x^{-1}$


Palindromic Cube with Non-Palindromic Root The only known palindromic cube with a root that is not itself palindromic is $10 \, 662 \, 526 \, 601$.


Restriction to Subset of Strict Total Ordering is Strict Total Ordering Let $S$ be a set or class. Let $\prec$ be a strict total ordering on $A$. Let $T$ be a subset or subclass of $A$. Then the restriction of $\prec$ to $B$ is a strict total ordering of $B$.


Division Theorem/Positive Divisor/Existence/Proof 1 For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$


Primitive of Power of x over a x squared plus b x plus c Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {x^m \rd x} {a x^2 + b x + c} = \frac {x^{m - 1} } {\paren {m - 1} a} - \frac b a \int \frac {x^{m - 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{m - 2} \rd x} {a x^2 + b x + c}$


Number does not divide Number iff Square does not divide Square Let $a, b \in \Z$ be integers. Then: :$a \nmid b \iff a^2 \nmid b^2$ where $a \nmid b$ denotes that $a$ is not a divisor of $b$. {{:Euclid:Proposition/VIII/16}}


Quaternion Group has Normal Subgroup without Complement Let $Q$ denote the quaternion group. There exists a normal subgroup of $Q$ which has no complement.


Sum of All Ring Products is Associative Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +}\right), \left({T, +}\right), \left({U, +}\right)$ be additive subgroups of $\left({R, +, \circ}\right)$. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\,n}\right]}\right\}$ Then: :$\left({S T}\right) U = S \left({T U}\right)$


Row Equivalence is Equivalence Relation Row equivalence is an equivalence relation.


Mappings in Product of Sets are Surjections/Family of Sets Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $\struct {P, \family {\phi_i} _{i \mathop \in I} }$ be a product of $S$ and $T$. Then for all $i \in I$, $\phi_i$ is a surjection.


Completion of Valued Field Let $\struct {k, \norm {\,\cdot\,} }$ be a valued field. Then there exists a completion $\struct {k', \norm {\,\cdot\,}'}$ of $\struct {k, \norm {\,\cdot\,} }$ such that $\struct {k', \norm {\,\cdot\,}'}$ is a valued field. Furthermore, every completion of $\struct{k, \norm {\,\cdot\,} }$ is isometrically isomorphic to $\struct {k', \norm {\,\cdot\,}'}$.


Absolute Value on Ordered Integral Domain is Strictly Positive except when Zero Let $\struct {D, +, \times, \le}$ be an ordered integral domain. For all $a \in D$, let $\size a$ denote the absolute value of $a$. Then $\size a$ is (strictly) positive except when $a = 0$.


Definite Integral from 0 to Pi of Sine of m x by Cosine of n x Let $m, n \in \Z$ be integers. Then: :$\displaystyle \int_0^\pi \sin m x \cos n x \rd x = \begin{cases} 0 & : m + n \text { even} \\ \dfrac {2 m} {m^2 - n^2} & : m + n \text { odd} \end{cases}$


Decomposition of Probability Measures Let $\struct {\Omega, \Sigma, P}$ be a probability space. Suppose that for every $\omega \in \Omega$, it holds that: :$\set \omega \in \Sigma$ that is, that $\Sigma$ contains all singletons. Then there exist a unique diffuse measure $\mu$ and a unique discrete measure $\nu$ such that: :$P = \mu + \nu$


Equivalence of Definitions of Weakly Locally Connected at Point Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. {{TFAE|def = Weakly Locally Connected at Point}}


Equivalence of Definitions of Abundant Number The following definitions of a abundant number are equivalent:


Closure is Closed Let $\struct {S, \preceq}$ be an ordered set. Let $\cl: S \to S$ be a closure operator. Let $x \in S$. Then $\map \cl x$ is a closed element of $S$ with respect to $\cl$.


Compact Complement Topology is Topology Let $T = \struct {\R, \tau}$ be the compact complement space. Then $\tau$ is a topology on $T$.


Characterisation of Local Rings Let $R$ be a ring. Let $J \lhd R$ be a maximal ideal. :$(1): \quad$ If the set $R \setminus J$ is precisely the group of units $R^\times$ of $R$, then $\tuple {R, J}$ is a local ring. :$(2): \quad$ If $1 + x$ is a unit in $R$ for all $x \in J$ then $\tuple {R, J}$ is local. {{explain|The specific interpretation of the notation $\tuple {R, J}$ is unclear. The usual notation for a ring is something like $\struct {R, +, \times}$.}}


Primitive of Cosecant of a x/Cosecant minus Cotangent Form :$\ds \int \csc a x \rd x = \frac 1 a \ln \size {\csc a x - \cot a x} + C$ where $\csc a x - \cot a x \ne 0$.


Rational Cut has Smallest Upper Number Let $r \in \Q$ be rational. Let $\alpha$ be the rational cut consisting of all rational numbers $p$ such that $p < r$. Then $\alpha$ is indeed a cut, and has a smallest upper number that is $r$.


Complex Numbers under Addition form Monoid The set of complex numbers under addition $\left({\C, +}\right)$ forms a monoid.


Integration on Polynomials is Linear Operator Let $P \left({\R}\right)$ be the vector space of all polynomial functions on the real number line $\R$. Let $S$ be the mapping defined as: :$\displaystyle \forall p \in P \left({\R}\right): \forall x \in \R: S \left({p \left({x}\right)}\right) = \int_0^x p \left({t}\right) \mathrm d t$ Then $S$ is a linear operator on $P \left({\R}\right)$.


Number whose Half is Odd is Even-Times Odd Let $a \in \Z$ be an integer such that $\dfrac a 2$ is an odd integer. Then $a$ is even-times odd. {{:Euclid:Proposition/IX/33}}


Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine :$\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$


First Order ODE/x dy - y dx = (1 + y^2) dy The first order ODE: :$(1): \quad x \rd y - y \rd x = \paren {1 + y^2} \rd y$ has the general solution: :$\dfrac x y = \dfrac 1 y - y + C$


Linear Subspace is Convex Set Let $V$ be a vector space over $\R$ or $\C$, and let $L$ be a linear subspace of $V$. Then $L$ is a convex set.


Squares of form 2 n^2 - 1 The sequence of integers $\left\langle{n}\right\rangle$ such that $2 n^2 - 1$ is square begins: :$1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, \ldots$ {{OEIS|A001653}}


Real Number Line with Point Removed is Not Path-Connected Let $\R$ be the real number line considered as an Euclidean space. Let $x \in \R$ be a real number. Then $\R \setminus \set x$, where $\setminus$ denotes set difference, is not path-connected.


Floor Function/Examples/Floor of Minus One Half :$\floor {-\dfrac 1 2} = -1$


Ring of Endomorphisms is not necessarily Commutative Ring Let $\struct {G, \oplus}$ be an abelian group. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $\struct {\mathbb G, \oplus, *}$ denote the '''ring of endomorphisms''' on $\struct {G, \oplus}$. Then $\struct {\mathbb G, \oplus, *}$ is not necessarily a commutative ring with unity.


Discrete Uniformity generates Discrete Topology Let $S$ be a set. Let $\UU$ be the discrete uniformity on $S$. Then the topology generated by $\UU$ is the discrete topology on $S$. The diagonal relation $\Delta_S$ generates the basis for this discrete topology.


Primitive of x squared over a x + b cubed :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^3} = \frac {2 b} {a^3 \paren {a x + b} } - \frac {b^2} {2 a^3 \paren {a x + b}^2} + \frac 1 {a^3} \ln \size {a x + b} + C$


Odd Order Complete Graph is Eulerian Let $K_n$ be the complete graph of $n$ vertices. Then $K_n$ is Eulerian {{iff}} $n$ is odd. If $n$ is even, then $K_n$ is traversable iff $n = 2$.


Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x Let $b \in \R$ be a real number. :$\forall x \in \R_{\ge 1}: \left\lfloor{\log_b x}\right\rfloor = \left\lfloor{\log_b \left\lfloor{x}\right\rfloor}\right\rfloor \iff b \in \Z_{> 1}$ where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.


Completely Irreducible Element equals Infimum of Subset implies Element Belongs to Subset Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $X \subseteq S$, $p \in S$ such that :$p$ is completely irreducible and $p = \inf X$ Then $p \in X$


Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17 Let $\struct {\Z_{16}, +}$ denote the additive group of integers modulo $16$. Let $\struct {\Z'_{17}, \times}$ denote the multiplicative group of reduced residues modulo $17$. Let $\phi: \struct {\Z_{16}, +} \to \struct {\Z'_{17}, \times}$ be the mapping defined as: :$\forall \eqclass k {16} \in \struct {\Z_{16}, +}: \map \phi {\eqclass k {16} } = \eqclass {3^k} {17}$ Then $\phi$ is a group isomorphism.