Completely Normal Space is Preserved under Homeomorphism If $T_A$ is a completely normal space, then so is $T_B$.


Open Set is G-Delta Set Let $T = \struct {S, \tau}$ be a topological space. Let $U$ be an open set of $T$. Then $U$ is a $G_\delta$ set of $T$.


Pairwise Independence does not imply Independence Just because all the events in a family of events in a probability space are pairwise independent, it does not mean that the family is independent.


Intersection of Subset with Upper Bounds Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $T^*$ be the set of all upper bounds of $T$ in $S$. Then $T^* \cap T \ne \varnothing$ {{iff}}: :$T$ has a greatest element $M$ and :$T^* \cap T$ is a singleton such that $T^* \cap T = \left\{{M}\right\}$


Power Set less Empty Set has no Smallest Element iff not Singleton Let $S$ be a set which is non-empty. Let $\mathcal C = \mathcal P \left({S}\right) \setminus \varnothing$, that is, the power set of $S$ without the empty set. Then the ordered structure $\left({\mathcal C, \subseteq}\right)$ has no smallest element {{iff}} $S$ is not a singleton.


Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers. Then $\rho$ is the rank function of a matroid on $S$ {{iff}} $\rho$ satisfies the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}}


Discrepancy between Gregorian Year and Tropical Year The Gregorian year and the tropical year differ such that the Gregorian calendar becomes $1$ day further out approximately every $3318$ years.


Combination Theorem for Sequences/Real/Quotient Rule/Corollary Let $\sequence {x_n}$ be convergent to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac 1 {x_n} = \frac 1 l$ provided that $l \ne 0$.


Equivalence of Definitions of Variance of Discrete Random Variable Let $X$ be a discrete random variable. Let $\mu = \expect X$ be the expectation of $X$. {{TFAE|def = Variance of Discrete Random Variable}}


333,667 is Only Prime whose Reciprocal is of Period 9 The only prime number whose reciprocal has a period of $9$ is $333 \, 667$: :$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$


Set of 3 Integers each Divisor of Sum of Other Two There exists exactly one set of distinct coprime positive integers such that each is a divisor of the sum of the other two: :$\set {1, 2, 3}$


Kernel of Composition of Ring Homomorphisms Let $f : A \to B$ and $g : B \to C$ be ring homomorphisms. Let $g \circ f$ be their composition. Then the kernel of $g \circ f$ is the preimage under $f$ of the kernel of $g$: :$\map \ker {g \circ f} = f^{-1} \sqbrk {\ker g}$


Domain of Real Square Function The domain of the real square function is the entire set of real numbers $\R$.


Element is Meet Irreducible iff Complement of Element is Irreducible Let $T = \struct {S, \tau}$ be a topological space. Let $P = \struct {\tau, \preceq}$ be an ordered set where $\mathord \preceq = \mathord \subseteq \cap \paren {\tau \times \tau}$ Let $A \in \tau$ such that :$A \ne \top_P$ where $\top_P$ denotes the greatest element in $P$. Then $A$ is meet irreducible in $P$ {{iff}} $\complement_S\left({A}\right)$ is irreducible where $\complement_S\left({A}\right)$ denotes the relative complement of $A$ relative to $S$.


121 is Square Number in All Bases greater than 2 Let $b \in \Z$ be an integer such that $b \ge 3$. Let $n$ be a positive integer which can be expressed in base $b$ as $121_b$. Then $n$ is a square number.


Null Measure is Measure Let $\struct {X, \Sigma}$ be a measurable space. Then the null measure $\mu$ on $\struct {X, \Sigma}$ is a measure.


Isomorphism between Group Generated by Reciprocal of z and 1 minus z and Symmetric Group on 3 Letters Let $S_3$ denote the symmetric group on $3$ letters. Let $G$ be the group generated by $1 / z$ and $1 - z$. Then $S_3$ and $G$ are isomorphic algebraic structures.


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


Multiplicative Inverse in Nicely Normed Algebra Let $A = \left({A_F, \oplus}\right)$ be a nicely normed $*$-algebra whose conjugation is denoted $*$. Let $a \in A$. Then the multiplicative inverse of $a$ is given by: :$a^{-1} = \dfrac {a^*} {\left \Vert {a}\right \Vert^2}$ where: : $a^*$ is the conjugate of $a$ : $\left \Vert {a}\right \Vert$ is the norm of $a$.


Ideal is Contained in Contraction of Extension Let $A$ and $B$ be commutative ring with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak a \subseteq A$ be an ideal. Then $\mathfrak a$ is contained in the contraction of its extension by $f$: :$\mathfrak a \subseteq \mathfrak a^{ec}$


Power Series Expansion of Reciprocal of Cube Root of 1 + x Let $x \in \R$ such that $-1 < x \le 1$. Then: :$\dfrac 1 {\sqrt [3] {1 + x} } = 1 - \dfrac 1 3 x + \dfrac {1 \times 4} {3 \times 6} x^2 - \dfrac {1 \times 4 \times 7} {3 \times 6 \times 9} x^3 + \cdots$


Second Order ODE/y'' - x f(x) y' + f(x) y = 0 The second order ODE: :$(1): \quad y'' - x \, \map f x y' + \map f x y = 0$ has the general solution: :$\displaystyle y = C_1 x + C_2 x \int x^{-2} e^{\int x \, \map f x \rd x} \rd x$


Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring/Corollary Let $\struct {R, \norm {\, \cdot \,} }$ be a complete normed division ring. Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$: :$x = \displaystyle \lim_{n \mathop \to \infty} x_n$


Subtraction has no Identity Element The operation of subtraction on numbers of any kind has no identity.


Compact Complement Space is not T2, T3, T4 or T5 Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is not a $T_2$ (Hausdorff) space, $T_{2 \frac 1 2}$ (completely Hausdorff) space, $T_3$ space, $T_4$ space or $T_5$ space.


Count of Commutative Binary Operations with Identity Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different commutative binary operations that can be applied to $S$ which have an identity element is given by: :$N = n^{\frac {n \left({n-1}\right)}2 + 1}$


Scalar Product with Inverse Unity :$\paren {-1_R} \circ x = - x$


Arctangent Function in terms of Gaussian Hypergeometric Function :$\displaystyle \arctan x = x \, {}_2 \map {F_1} {\frac 1 2, 1; \frac 3 2; -x^2}$


Field is Galois over Fixed Field of Automorphism Group Let $E/F$ be a finite field extension. Let $K = \operatorname{Fix}_E(\operatorname{Aut}(E/F))$ be the fixed field of the automorphism group of $E/F$. Then $E/K$ is Galois.


Permutation Representation defines Group Action Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\map \Gamma X$ be the symmetric group of $X$. Let $\rho: G \to \map \Gamma X$ be a permutation representation, that is, a homomorphism. The mapping $\phi: G \times X \to X$ associated to $\rho$, defined by: :$\map \phi {g, x} = \map {\map \rho g} x$ is a group action.


Arithmetic Sequence of 16 Primes The $16$ integers in arithmetic sequence defined as: :$2\,236\,133\,941 + 223\,092\,870 n$ are prime for $n = 0, 1, \ldots, 15$.


Relation of Boubaker Polynomials to Dickson Polynomials The Boubaker polynomials $B_n$ are linked to the  Dickson polynomials by the relations: : $B_{n+1} \left({x}\right) B_{n+j} \left({x}\right) - B_{n+j+1} \left({x}\right) B_n \left({x}\right) = \left({3 x^2 + 4}\right) D_{n+1} \left({x, \dfrac 1 4}\right)$ : $B_n \left({x}\right) = D_n \left({2x, \dfrac 1 4}\right) + 4 D_{n-1} \left({2 x, \dfrac 1 4}\right)$


Cauchy Sequences in Vector Spaces with Equivalent Norms Coincide Let $M_a = \struct {X, \norm {\, \cdot \, }_a}$ and $M_b = \struct {X, \norm {\, \cdot \,}_b}$ be normed vector spaces. Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $M_a$. Suppose, $\norm {\, \cdot \, }_a$ and $\norm {\, \cdot \,}_b$ are equivalent norms, i.e. $\norm {\, \cdot \, }_a \sim \norm {\, \cdot \,}_b$. Then $\sequence {x_n}_{n \mathop \in \N}$ is also a Cauchy sequence in $M_b$.


Order Topology equals Dual Order Topology Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $\tau$ be the $\preceq$-order topology on $S$. Let $\tau'$ be the $\succeq$-order topology on $S$, where $\succeq$ is the dual ordering of $\preceq$. Then $\tau' = \tau$.


Max is Half of Sum Plus Absolute Difference For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$: :$\max \set {a, b} = \frac 1 2 \paren {a + b + \size {a - b} }$


Summation over k of Floor of k over 2 :$\displaystyle \sum_{k \mathop = 1}^n \floor {\dfrac k 2} = \floor {\dfrac {n^2} 4}$


Prime to Own Power minus 1 over Prime minus 1 being Prime Let $n \in \Z_{>1}$ be an integer greater than $1$. Then $\dfrac {n^n - 1} {n - 1}$ is a prime for $n$ equal to: :$2, 3, 19, 31$ {{OEIS|A088790}}


Singleton Set is not Dense-in-itself Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Then the singleton set $\left\{{x}\right\}$ is not dense-in-itself.


Limit to Infinity of Complementary Error Function :$\displaystyle \lim_{x \mathop \to \infty} \map \erfc x = 0$


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


Dimension of Vector Space of Polynomial Functions Let $\struct {F, +, \times}$ be a field whose unity is $1_F$. Let $F_n \sqbrk X$ be the ring of polynomials over $F$ whose degree is less than $n$. Then the dimension of the vector space $F_n \sqbrk X$ is $n$.


Laplace Transform of t^2 by Cosine a t Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {t^2 \cos a t} = \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}$


Area of Integer Heronian Triangle is Multiple of 6 Let $\triangle {ABC}$ be an integer Heronian triangle. Then the area of $\triangle {ABC}$ is a multiple of $6$.


Expansion of Included Set Topology Let $S$ be a set. Let $A_1 \subseteq S$ and $A_2 \subseteq S$. Let $T_1 = \struct {S, \tau_{A_1} }$ and $T_2 = \struct {S, \tau_{A_2} }$ be included set spaces on $S$. Then: :$(1): \quad T_1 \ge T_2 \iff A_1 \subseteq A_2$ :$(2): \quad T_1 > T_2 \iff A_1 \subsetneq A_2$ where: :$T_1 \ge T_2$ denotes that $T_1$ is finer than $T_2$ :$T_1 > T_2$ denotes that $T_1$ is strictly finer than $T_2$.


Semilattice Homomorphism is Order-Preserving Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a semilattice homomorphism. Let $\preceq_1$ be the ordering on $S$ defined by: :$a \preceq_1 b \iff \paren {a \circ b} = b$ Let $\preceq_2$ be the ordering on $T$ defined by: :$x \preceq_2 y \iff \paren {x * y} = y$ Then: :$\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ is order-preserving


D'Alembert's Formula {{MissingLinks|throughout}} Let $u: \R^2 \to \R$ be a twice-differentiable function in two variables.  Let $\phi: \R \to \R$ be a differentiable function in $x$.  Let $\psi: \R \to \R$ be an integrable function in $x$. {{explain|What does "in $x$" mean? $x$ has not been defined yet.}} Let $c \in \R_{> 0}$ be a constant. Then the solution to the partial differential equation: :$u_{tt} = c^2 u_{xx}$ {{explain|The condensed form of the partial differential equation is not explicit enough, as neither $x$ nor $t$ have been defined. I refer you to Definition:Mapping/Notation/Warning where it is explained that, say, you can't say "Let $\map u {x, t}$ be a function". Recommend that something like $\dfrac {\partial^2} {\partial x^2} \map u {x, t} {{=}} c^2 \dfrac {\partial^2} {\partial t^2} \map u {x, t}$ be used.}} with initial conditions {{begin-eqn}} {{eqn | l = \map u {x, 0}       | r = \map \phi x }} {{eqn | l = \map {u_t} {x, 0}       | r = \map \psi x }} {{end-eqn}} is given by: :$\displaystyle \map u {x, t} = \dfrac 1 2 \paren {\map \phi {x + c t} + \map \phi {x - c t} } + \dfrac 1 {2 c} \int_{x - c t}^{x + c t} \map \psi s \rd s$ The above solution formula is called d'Alembert's Formula.


Equivalence of Definitions of Cosine of Angle Let $\theta$ be an angle. {{TFAE|def = Cosine of Angle|view = cosine}}


Primitive of Reciprocal of x fourth minus a fourth :$\displaystyle \int \frac {\d x} {x^4 - a^4} = \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a$


Product of Number of Edges, Edges per Face and Faces of Regular Dodecahedron The product of the number of edges, edges per face and faces of a regular dodecahedron is $1800$.


Sum of 2 Squares in 2 Distinct Ways/Examples/65 $65$ can be expressed as the sum of two square numbers in two distinct ways: {{begin-eqn}} {{eqn | l = 65       | r = 8^2 + 1^2 }} {{eqn | r = 7^2 + 4^2 }} {{end-eqn}}


4 Consecutive Integers cannot be Square-Free Let $n, n + 1, n + 2, n + 3$ be four consecutive positive integers. At least one of these is not square-free.


Numbers Equal to Sum of Squares of Digits There are exactly $2$ integers which are equal to the sum of the squares of their digits when expressed in base $10$: :$0 = 0^2$ :$1 = 1^2$


Absorption Laws (Set Theory)/Corollary :$S \cup \paren {S \cap T} = S \cap \paren {S \cup T}$


Card Game with Bluffing is Completely Mixed Game The game of cards with bluffing is a completely mixed game.


Rational Numbers whose Denominators are not Divisible by 4 do not form Ring Let $S$ be the set defined as: :$S = \set {\dfrac m n : m, n \in \Z, m \perp n, 4 \nmid n}$ That is, $S$ is defined as the set of rational numbers such that, when expressed in canonical form, their denominators are not divisible by $4$. Then the algebraic structure $\struct {S, +, \times}$ is not a ring.


Real Numbers under Addition form Monoid The set of real numbers under addition $\struct {\R, +}$ forms a monoid.


If Ratio of Cube to Number is as between Two Cubes then Number is Cube Let $a, b, c, d \in \Z$ be integers such that: :$\dfrac a b = \dfrac {c^3} {d^3}$ Let $a$ be a cube number. Then $b$ is also a cube number. {{:Euclid:Proposition/VIII/25}}


Condition for Elements of Group to be in Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x, y \in G$ be such that $2$ elements of $\set {x, y, x y}$ are elements of $h$. Then ''all'' the elements of $\set {x, y, x y}$ are in $H$.


Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations {{TFAE|def = Semantic Equivalence for Boolean Interpretations}} Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.


Real Number Line is Separable Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is separable.


Loop-Digraph as a Relation A loop-digraph is the same thing as a relational structure.


Sum of Sequence of Cubes divides 3 times Sum of Sequence of Fifth Powers :$\displaystyle \sum_{i \mathop = 1}^n i^3 \divides 3 \sum_{i \mathop = 1}^n i^5$ where $\divides$ denotes divisibility.


Parseval's Theorem/Formulation 2 Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$. Let $f$ be expressed by the Fourier series: :$\map f x = \displaystyle \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$ where: :$c_n = \displaystyle \frac 1 {2 \pi} \int_{-\pi}^\pi \map f t e^{-i n t} \rd t$ Then: :$\displaystyle \frac 1 {2 \pi} \int_{-\pi}^\pi \size {\map f x}^2 \rd x = \sum_{n \mathop = -\infty}^\infty \size {c_n}^2$


Euler Phi Function of 1 :$\map \phi 1 = 1$


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


Image of Set Difference under Mapping/Corollary 2 Let $f: S \to T$ be a mapping. Let $X$ be a subset of $S$. Then: :$\relcomp {\Img f} {f \sqbrk X} \subseteq f \sqbrk {\relcomp S X}$ where: :$\Img f$ denotes the image of $f$ :$\complement_{\Img f}$ denotes the complement relative to $\Img f$. This can be expressed in the language and notation of direct image mappings as: :$\forall X \in \powerset S: \relcomp {\Img f} {\map {f^\to} X} \subseteq \map {f^\to} {\relcomp S X}$ That is: :$\forall X \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } X \subseteq \map {\paren {f^\to \circ \complement_S} } X$ where $\circ$ denotes composition of mappings.


Category of Finite Sets is Category Let $\mathbf{Set}$ be the category of finite sets. Then $\mathbf{Set}$ is a metacategory.


Triangle Inequality for Generalized Sums Let $V$ be a Banach space, and let $\left\Vert{\cdot}\right\Vert$ denote its norm. Let $\left({v_i}\right)_{i\in I}$ be an indexed subset of $V$. Let the generalized sum $\displaystyle \sum \left\{{v_i: i \in I}\right\}$ converge absolutely. Then $\displaystyle \left\Vert{\sum \left\{{v_i: i \in I}\right\}}\right\Vert \le \sum \left\{{\left\Vert{v_i}\right\Vert: i \in I}\right\}$.


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


Parallel Law for Extremal Length Let $X$ be a Riemann surface. Let $\Gamma_1, \Gamma_2$ be families of rectifiable curves (or, more generally, families of disjoint unions of rectifiable curves) on $X$. Let $\Gamma_1$ and $\Gamma_2$ be disjoint, in the sense that: :there exist disjoint Borel subsets $A_1, A_2 \subseteq X$ such that: ::for any $\gamma_1 \in \Gamma_1$ and $\gamma_2 \in \Gamma_2$, we have $\gamma_1 \subseteq A_1$ and $\gamma_2 \subseteq A_2$. Let $\Gamma$ be a third curve family, with the property that every element $\Gamma_1$ and every element of $\Gamma_2$ contains some element of $\Gamma$. Then the extremal length of $\Gamma$ satisfies: :$\dfrac 1 {\lambda \left({\Gamma}\right)} \ge \dfrac 1 {\lambda \left({\Gamma_1}\right)} + \dfrac 1 {\lambda \left({\Gamma_2}\right)}$


Product of Coprime Ideals equals Intersection Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be coprime ideals. Then their product equals their intersection: :$\mathfrak a \mathfrak b = \mathfrak a \cap \mathfrak b$


Union of Blocks is Set of Points Let $\struct {X, \mathcal B}$ be a pairwise balanced design. That is, let $\struct {X, \mathcal B}$ be a design, with $\size X \ge 2$, and the number of occurrences of each pair of distinct points in $\mathcal B$ be $\lambda$ for some $\lambda > 0$ constant. Then the set union of all the subset elements in $\mathcal B$ is precisely $X$.


Linear Combination of Sequence is Linear Combination of Set Let $G$ be an $R$-module. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$. Let $b$ be an element of $G$. Then: :$b$ is a linear combination of the sequence $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ {{iff}}: :$b$ is a linear combination of the set $\set {a_k: 1 \mathop \le k \mathop \le n}$


Linear Transformations form Abelian Group Let $\struct {G, +_G}$ and $\struct {H, +_H}$ be groups. Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules. Let $\map {\mathcal L_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Let $\oplus_H$ be the operation on $H^G$ as defined in Addition of Linear Transformations. Then $\struct {\map {\LL_R} {G, H}, \oplus_H}$ is an abelian group.


Finite Product Space is Connected iff Factors are Connected/General Case Let $I$ be an indexing set. Let $\family {T_\alpha}_{\alpha \mathop \in I}$ be an indexed family of topological spaces. Let $T = \displaystyle \prod_{\alpha \mathop \in I} T_\alpha$ be the Cartesian space of $\family {T_\alpha}_{\alpha \mathop \in I}$. Let $T = \displaystyle \overline {\bigcup_{\alpha \mathop \in I} S_\alpha}$. {{explain|Notation in the above -- explain what the overline means in this context.}} Let $\tau$ be a topology on $T$ such that the subsets ${S'}_\alpha \subseteq \displaystyle \prod T_\alpha$ where ${S'}_\alpha = \set {\family {y_\beta} \in T: y_\beta = x \beta \text { for all } \beta \ge \alpha}$ is homeomorphic to $S_{\alpha - 1} \times T_\alpha$. Then $T$ is connected {{iff}} each of $T_\alpha: \alpha \in I$ are connected.


First Order ODE/y' = (x + y)^2 The first order ODE: :$\dfrac {\d y} {\d x} = \paren {x + y}^2$ has the general solution: :$x + y = \map \tan {x + C}$


Intersection of Coprime Cyclic Subgroups is Trivial Let $G$ be a group whose identity is $e$. Let $x, y \in G$ such that: :$\order x \perp \order y$ where: :$\order x, \order y$ denotes the orders of $x$ and $y$ in $G$ respectively :$\perp$ denotes the coprimality relation. Then: :$\gen x \cap \gen y = \set e$ where $\gen x, \gen y$ denotes the subgroup of $G$ generated by $x$ and $y$ in $G$ respectively.


Current in Electric Circuit/L, R in Series/Exponentially Decaying EMF at t = 0 Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let an EMF $E$ be imposed upon $K$ at time $t = 0$ defined by the equation: :$E = E_0 e^{-k t}$ The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} {R - k L} e^{-k t} + \left({I_0 - \dfrac {E_0} {R - k L} }\right) e^{-R t / L}$


11 is Only Palindromic Prime with Even Number of Digits $11$ is the only palindromic prime with an even number of digits when expressed in decimal notation.


Fourier's Theorem/Integral Form/Continuous Point Let $f$ be continuous at $t \in \R$. Then: :$\displaystyle \map f t = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i d t} \map f t \rd t} \rd s$


Product of Absolutely Continuous Functions is Absolutely Continuous Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be absolutely continuous functions. Then $f \times g$ is absolutely continuous.


Cartesian Product with Complement Let $S$ and $T$ be sets. Let $A \subseteq S$ and $B \subseteq T$ be subsets of $S$ and $T$, respectively. Let $\relcomp S A$ denote the relative complement of $A$ in $S$. Then: {{begin-eqn}} {{eqn | l = \relcomp S A \times T        | r = \relcomp {S \times T} {A \times T} }} {{eqn | l = S \times \relcomp T B       | r = \relcomp {S \times T} {S \times B} }} {{end-eqn}}


Existence of Prime between Prime and Factorial Let $p$ be a prime number. Then there exists a prime number $q$ such that: :$p < q \le p! + 1$ where $p!$ denotes the factorial of $p$.


Rank of Ordinal Let $x$ be an ordinal. Let $\map {\operatorname {rank} } x$ denote the rank of $x$. Then: :$\map {\operatorname {rank} } x = x$


Arens-Fort Space is Paracompact Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a paracompact space.


Set Products on Same Set are Equivalent Let $S$ and $T$ be sets. Let $\struct {P, \phi_1, \phi_2}$ and $\struct {Q, \psi_1, \psi_2}$ be products of $S$ and $T$. Then there exists a unique bijection $\chi: Q \to P$ such that: :$\phi_1 \circ \chi = \psi_1$ :$\phi_2 \circ \chi = \psi_2$


Nicely Normed Alternative Algebra is Normed Division Algebra $A = \left({A_F, \oplus}\right)$ be a nicely normed $*$-algebra which is also an alternative algebra. Then $A$ is a normed division algebra.


Hensel's Lemma for Composite Numbers Let $b \in \Z \setminus \set {-1, 0, 1}$ be an integer. Let $k > 0$ be a positive integer. Let $\map f X \in \Z \sqbrk X$ be a polynomial. Let $x_k \in \Z$ such that: :$\map f {x_k} \equiv 0 \pmod {b^k}$ :$\gcd \set {\map {f'} {x_k}, b} = 1$ Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such that: :$\map  f {x_{k + l} } \equiv 0 \pmod {b^{k + l} }$ :$x_{k + l} \equiv x_k \pmod {b^k}$ and any two integers satisfying these congruences are congruent modulo $b^{k + l}$. Moreover, for all $l \ge 0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$: :$x_{k + l + 1} \equiv x_{k + l} - \dfrac {\map f {x_{k + l} } } {\map {f'} {x_{k + l} } } \pmod {b^{k + l + 1} }$ :$x_{k + l + 1} \equiv x_{k + l} \pmod {b^{k + l} }$


Product with Inverse equals Identity iff Equality Let $\struct {G, \circ}$ be a group whose identity element is $e$. Then: :$\forall a, b \in G: a \circ b^{-1} = e \iff a = b$


Definite Integral from 0 to Pi of x by Logarithm of Sine x :$\displaystyle \int_0^\pi x \, \map \ln {\sin x} \rd x = -\frac {\pi^2} 2 \ln 2$


Uncountable Closed Ordinal Space is Sigma-Compact Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is a $\sigma$-compact space.


Full Rook Matrix is Invertible A full rook matrix is invertible.


Equivalence of Definitions of Maximal Element Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. {{TFAE|def = Maximal Element}}


Set of Natural Numbers is Primitive Recursive The set of natural numbers $\N$ is primitive recursive.


Doubleton of Sets can be Derived using Comprehension Principle Let $a$ and $b$ be sets. By application of the comprehension principle, the set $\set {a, b}$ can be formed. Hence the '''doubleton''' $\set {a, b}$ can be derived as a valid object in Frege set theory.


Complex Sine Function is Unbounded Let $\sin: \C \to \C$ be the complex sine function.  Then $\sin$ is unbounded.


Condition for Divisibility of Powers of Prime Let $p$ be a prime. Let $k, l \in \Z_{>0}$. Then: :$p^k \divides p^l \iff k \le l$


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


Combination Theorem for Cauchy Sequences/Combined Sum Rule :$\sequence {a x_n + b y_n }$ is a Cauchy sequence.


Separation Properties Preserved in Subspace/Corollary If $T$ has one of the following properties then $T_H$ has the same property: :Regular Property :Tychonoff (Completely Regular) Property :Completely Normal Property That is, the above properties are all hereditary.


Boundary of Union of Separated Sets equals Union of Boundaries Let $T$ be a topological space. Let $A, B$ be subsets of $T$. Let $A$ and $B$ are separated. Then: :$\partial \left({ A \cup B }\right) = \partial A \cup \partial B$ where : $\partial A$ denotes the boundary of $A$, : $A \cup B$ denotes the union of $A$ and $B$.


Product of Number of Edges, Edges per Face and Faces of Cube The product of the number of edges, edges per face and faces of a cube is $288$.


Definition:Smith-Volterra-Cantor Set Let $G$ be a Cantor collection. Let $g_0 \in G$ such that $\map \mu {g_0} = b$. Let $p$ be a natural number. Then there are two nonempty disjoint sets $N_p$ and $P$ such that: :$g_0 = N_p \bigcup P$ where $N_p$ is nowhere dense in the relative topology on $g_0$ and $\map \mu P \le b / p$.


Existence of Strongly Locally Compact Space which is not Weakly Sigma-Locally Compact There exists at least one example of a strongly locally compact topological space which is not also a weakly $\sigma$-locally compact space.


Characteristic of Finite Ring is Non-Zero Let $\struct {R, +, \circ}$ be a finite ring with unity. Then the characteristic of $R$ is not zero.


First Order ODE/y - x y' = y' y^2 exp y The first order ODE: :$(1): \quad y - x y' = y' y^2 e^y$ has the general solution: :$x y^2 = e^y + C$


Space of Bounded Sequences with Supremum Norm forms Banach Space Let $\ell^\infty$ be a space of bounded sequences. Let $\norm {\, \cdot \,}_\infty$ be a supremum norm. Then $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is a Banach space.


Order Embedding into Image is Isomorphism Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $S'$ be the image of a mapping $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$. Then: :$\phi$ is an order embedding from $\struct {S, \preceq_1}$ into $\struct {T, \preceq_2}$ {{iff}}: :$\phi$ is an order isomorphism from $\struct {S, \preceq_1}$ into $\struct {S', \preceq_2 \restriction_{S'} }$.


Carmichael Number has 3 Odd Prime Factors Let $n$ be a Carmichael number. Then $n$ has at least $3$ distinct odd prime factors.


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


Discrete Space is Separable iff Countable Let $T = \left({S, \tau}\right)$ be a discrete topological space. Then: :$T$ is separable {{iff}} $S$ is countable.


T1/2 Space is T0 Space Let $T = \struct {S, \tau}$ be a $T_{\frac 1 2}$ topological space. Then $T$ is $T_0$ space.


Pi as Sum of Sequence of Reciprocal of Product of Three Consecutive Integers :$\dfrac {\pi - 3} 4 = \dfrac 1 {2 \times 3 \times 4} - \dfrac 1 {4 \times 5 \times 6} + \dfrac 1 {6 \times 7 \times 8} \cdots$


Sum of Binomial Coefficients over Lower Index/Corollary :$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{i \mathop \in \Z} \binom n i = 2^n$


Chebyshev Distance is Limit of P-Product Metric Let $M_{1'} = \left({A_{1'}, d_{1'}}\right)$ and $M_{2'} = \left({A_{2'}, d_{2'}}\right)$ be metric spaces. Let $\mathcal A = A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$. Let $p \in \R_{\ge 1}$. Let $d_p: \mathcal A \times \mathcal A \to \R$ be the $p$-product metric on $\mathcal A$ : $d_p \left({x, y}\right) := \left({\left({d_{1'} \left({x_1, y_1}\right)}\right)^p + \left({d_{2'} \left({x_2, y_2}\right)}\right)^p}\right)^{1/p}$ and let $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) := \max \left\{ {d_{1'} \left({x_1, y_1}\right), d_{2'} \left({x_2, y_2}\right)}\right\}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \mathcal A$. Then: :$\displaystyle d_\infty = \lim_{p \mathop \to \infty} d_p$ in the sense that: :$\displaystyle \max \left\{ {d_{1'} \left({x_1, y_1}\right), d_{2'} \left({x_2, y_2}\right)}\right\} = \lim_{p \mathop \to \infty} \left({\left({d_{1'} \left({x_1, y_1}\right)}\right)^p + \left({d_{2'} \left({x_2, y_2}\right)}\right)^p}\right)^{1/p}$


Accumulation Point of Sequence is not necessarily Limit Point Let $T = \struct {S, \tau}$ be a topological space. Let $\sequence {a_n}$ be a sequence in $T$. Let $q \in S$ be an accumulation point of $\sequence {a_n}$. Then it is not necessarily the case that $q$ is also a limit point of $\sequence {a_n}$.


Derivative of General Logarithm of Function :$\map {\dfrac \d {\d x} } {\log_a u} = \dfrac {\log_a e} u \dfrac {\d u} {\d x}$


Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares The largest odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers with greatest common divisor $1$ is $157$. The full sequence of such odd positive integers which cannot be so expressed is: :$1, \ldots, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 67, 69, 73, 77, 83, 89, 97, 101, 103, 115, 157$ where the sequence contains all odd integers between $1$ and $37$. This sequence appears not to be documented on the {{OEISLink}}.


23 is Largest Integer not Sum of Distinct Perfect Powers The largest integer which cannot be expressed as the sum of distinct perfect powers is $23$.


Topology as Magma of Sets The concept of a topology is an instance of a magma of sets.


Fourth Isomorphism Theorem Let $\phi: R \to S$ be a ring homomorphism. Let $K = \map \ker \phi$ be the kernel of $\phi$. Let $\mathbb K$ be the set of all subrings of $R$ which contain $K$ as a subset. Let $\mathbb S$ be the set of all subrings of $\Img \phi$. Let $\phi^\to: \powerset R \to \powerset S$ be the direct image mapping of $\phi$. Then its restriction $\phi^\to: \mathbb K \to \mathbb S$ is a bijection. Also: :$(1): \quad \phi^\to$ and its inverse both preserve subsets. :$(2): \quad \phi^\to$ and its inverse both preserve ideals: :::: If $J$ is an ideal of $R$, then $\map {\phi^\to} J$ is an ideal of $S$ :::: If $J'$ is an ideal of $S$, then $\map {\paren {\phi^\to}^{-1} } {J'}$ is an ideal of $R$


Integral Operator is Linear/Corollary 1 :$\forall \alpha, \beta \in \R: \map T {f + g} = \map T f + \map T g$


Product of Negative Real Numbers is Positive Let $a, b \in \R_{\le 0}$ be negative real numbers. Then: :$a \times b \in \R_{\ge 0}$ That is, their product $a \times b$ is a positive real number.


Recursive Construction of Transitive Closure Given the relation $\mathcal R$, its transitive closure $\mathcal R^+$ can be constructed as follows: Let: :$\mathcal R_n := \begin{cases} \mathcal R & : n = 0 \\ \mathcal R_{n-1} \cup \left\{{\left({x_1, x_3}\right): \exists x_2: \left({x_1, x_2}\right) \in \mathcal R_{n-1} \land \left({x_2, x_3}\right) \in \mathcal R_{n-1}}\right\} & : n > 0 \end{cases}$ Finally, let: :$\displaystyle \mathcal R^+ = \bigcup_{i \in \N} \mathcal R_i$. Then $R^+$ is the transitive closure of $R$.


Canonical Injection of Real Number Line into Complex Plane Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $f: \R \to \C$ be the mapping from the real numbers to the complex numbers defined as: :$\forall x \in \R: \map f z = x + 0 y$ Then $f: \struct {\R, +} \to \struct {\C, +}$ is a monomorphism.


Sum over k to n of Unsigned Stirling Number of the First Kind of k with m by n factorial over k factorial Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$ where: :$\displaystyle {k \brack m}$ denotes an unsigned Stirling number of the first kind :$ n!$ denotes a factorial.


Complex Numbers under Addition form Group Let $\C$ be the set of complex numbers. The structure $\struct {\C, +}$ is a group.


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


Sum of Infinite Arithmetic-Geometric Sequence Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as: :$a_k = \paren {a + k d} r^k$ for $n = 0, 1, 2, \ldots$ Let: :$\size r < 1$ where $\size r$ denotes the absolute value of $r$. Then: :$\displaystyle \sum_{n \mathop = 0}^\infty \paren {a + k d} r^k = \frac a {1 - r} + \frac {r d} {\paren {1 - r}^2}$


Odd Order Group Element is Square Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $x \in G$. Then: :$\exists y \in G: y^2 = x$ {{iff}}: :the order  $\order x$ is odd


Composition of Permutations is not Commutative Let $S$ be a set. Let $\map \Gamma S$ denote the set of permutations on $S$. Let $\pi, \rho$ be elements of $\map \Gamma S$ Then it is not necessarily the case that: :$\pi \circ \rho = \rho \circ \pi$ where $\circ$ denotes composition.


Series Expansion for Pi over 8 Root 2 :$\displaystyle \frac \pi {8 \sqrt 2} = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {2 n - 1} {\paren {4 n - 1} \paren {4 n - 3} }$


Sum of Powers of 2 Let $n \in \N_{>0}$ be a (strictly positive) natural number. Then: {{begin-eqn}} {{eqn | n = 1       | l = 2^n - 1       | r = \sum_{j \mathop = 0}^{n - 1} 2^j       | c =  }} {{eqn | r = 1 + 2 + 2^2 + 2^3 + \dotsb + 2^{n - 1}       | c =  }} {{end-eqn}}


Divergent Sequence may be Bounded While every Convergent Sequence is Bounded, it does not follow that every bounded sequence is convergent. That is, there exist bounded sequences which are divergent.


Smallest Field is Field The ring $\struct {\set {0_R, 1_R}, +, \circ}$ is the smallest algebraic structure which is a field.


Derivative of Inverse Hyperbolic Secant Function :$\map {\dfrac \d {\d x} } {\sech^{-1} u} = \dfrac {-1} {u \sqrt {1 - u^2} } \dfrac {\d u} {\d x}$  where $0 < u < 1$


Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in a vector space of $3$ dimensions: Let $\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right)$ denote the scalar triple product of $\mathbf a, \mathbf b, \mathbf c$. Then $\left\lvert{\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right)}\right\rvert$ equals the volume of the parallelepiped contained by $\mathbf a, \mathbf b, \mathbf c$.


Pythagorean Triangles whose Area equal their Perimeter There exist exactly $2$ Pythagorean triples which define a Pythagorean triangle whose area equals its perimeter: :$(1): \quad \tuple {6, 8, 10}$, leading to an area and perimeter of $24$ :$(2): \quad \tuple {5, 12, 13}$, leading to an area and perimeter of $30$.


Condition for Well-Foundedness/Reverse Implication/Proof 2 Let $\struct {S, \preceq}$ be an ordered set. Suppose that there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that: :$\forall n \in \N: a_{n + 1} \prec a_n$ Then $\struct {S, \preceq}$ is well-founded.


Integer as Sum of 27 Primes Every positive integer greater than $1$ can be expressed as the sum of no more than $27$ primes.


Factorial is Primitive Recursive The factorial function $\operatorname{fac}: \N \to \N$ defined as: :$\operatorname{fac} \left({n}\right) = n!$ is primitive recursive.


Left Regular Representation is Transitive Group Action Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $*: G \times G \to G$ be the group action: :$\forall g, h \in G: g * h = \map {\lambda_g} h$ where $\lambda_g$ is the left regular representation of $G$ with respect to $g$. Then $*$ is a transitive group action.


Arc of Cycloid is Concave Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. Then the locus of $P$ is concave.


Equivalence of Definitions of Infimum of Real-Valued Function Let $S \subseteq \R$ be a subset of the real numbers. Let $f: S \to \R$ be a real function on $S$. {{TFAE|def = Infimum of Real-Valued Function}}


Multiply Perfect Number of Order 8 The number defined as: :$n = 2^{65} \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29^2 \times 31^2$ ::$\times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83 \times 89 \times 103 \times 127 \times 131$ ::$\times 149 \times 211 \times 307 \times 331 \times 463 \times 521 \times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113$ ::$\times 6481 \times 10 \, 429 \times 20 \, 857 \times 110 \, 563 \times 599 \, 479 \times 16 \, 148 \, 168 \, 401$ is multiply perfect of order $8$.


Wilson's Theorem/Corollary 1 Let $p$ be a prime number. Then $p$ is the smallest prime number which divides $\paren {p - 1}! + 1$.


Moment Generating Function of Discrete Uniform Distribution Let $X$ be a discrete random variable with a discrete uniform distribution with parameter $n$ for some $n \in \N$. Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = \dfrac {e^t \paren {1 - e^{n t} } } {n \paren {1 - e^t} }$


Order Topology is Normal Let $\struct {S, \preceq}$ be a toset. Let $\tau$ be the order topology on $S$. Then $\struct {S, \tau}$ is normal.


Binomial Coefficient is instance of Gaussian Binomial Coefficient Let $\dbinom r m_q$ denote the Gaussian binomial coefficient: Then: :$\displaystyle \lim_{q \mathop \to 1^-} \dbinom r m_q = \dbinom r m$ where $\dbinom r m$ denotes the conventional binomial coefficient.


Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets Let $S$ and $T$ be disjoint sets. Let $\powerset S$ denote the power set of $S$. Then there exists a bijection between $\powerset {S \cup T}$ and $\paren {\powerset S} \times \paren {\powerset T}$. Hence: :$\powerset {S \cup T} \sim \paren {\powerset S} \times \paren {\powerset T}$ where $\sim$ denotes set equivalence.


Category has Finite Limits iff Finite Products and Equalizers Let $\mathbf C$ be a metacategory. Then: :$\mathbf C$ has all finite limits {{iff}}: :$\mathbf C$ has all finite products and equalizers.


Primitive of x squared by Inverse Hyperbolic Cotangent of x over a :$\displaystyle \int x^2 \coth^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \coth^{-1} \frac x a + \frac {a^3} 6 \ln \left({x^2 - a^2}\right) + C$


Minimal Uncountable Well-Ordered Set Unique up to Isomorphism Let $\Omega, \Omega'$ be minimal uncountable well-ordered sets. Then $\Omega$ is order isomorphic to $\Omega'$. That is, the minimal uncountable well-ordered set is unique up to order isomorphism.


Scalar Multiplication Corresponds to Multiplication by 1x1 Matrix Let $\map \MM 1$ denote the matrix space of square matrices of order $1$. Let $\map \MM {1, n}$ denote the matrix space of order $1 \times n$. Let $\mathbf A = \begin {pmatrix} a \end {pmatrix} \in \map \MM 1$ and $\mathbf B = \begin {pmatrix} b_1 & b_2 & \cdots & b_n \end{pmatrix} \in \map \MM {1, n}$. Let $\mathbf C = \mathbf A \mathbf B$ denote the (conventional) matrix product of $\mathbf A$ with $\mathbf B$. Let $\mathbf D = a \mathbf B$ denote the matrix scalar product of $a$ with $\mathbf B$. Then $\mathbf C = \mathbf D$.


Superabundant Numbers are Infinite in Number There are infinitely many superabundant numbers.


Sequence of 9 Primes of form 4n+1 The following sequence of $9$ consecutive prime numbers are all of the form $4 n + 1$: :$11 \, 593, 11 \, 597, 11 \, 617, 11 \, 621, 11 \, 633, 11 \, 657, 11 \, 677, 11 \, 681, 11 \, 689$


Set of Rationals Less than Root 2 has no Greatest Element Let $A$ be the set of all positive rational numbers $p$ such that $p^2 < 2$. Then $A$ has no greatest element.


Different Representations to Number Base represent Different Integers Let $k \in \Z$ such that $k \ge 2$. Let $a$ and $b$ be representations of integers in base $k$ notation: :$a = \displaystyle \sum_{j \mathop = 0}^r a_j k^j$ :$b = \displaystyle \sum_{j \mathop = 0}^s b_j k^j$ such that either: :$r \ne s$ or: :$\exists j \in \set {0, 1, \ldots, r}: a_j \ne b_j$ Then $a$ and $b$ represent different integers.


Subset not necessarily Submagma Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$. Then it is not necessarily the case that: : $\struct {T, \circ} \subseteq \struct {S, \circ}$ That is, it does not always follow that $\struct {T, \circ}$ is a submagma of $\struct {S, \circ}$.


Set of Common Divisors of Integers is not Empty Let $a, b \in \Z$ be integers. Let $S$ be the set of common divisors of $a$ and $b$. Then $S$ is not empty.


Torelli's Sum :$\displaystyle \left({x + y}\right)^{\overline n} = \sum_k \binom n k x \left({x - k z + 1}\right)^{\overline {k - 1} } \left({y + k z}\right)^{\overline {n - k} }$ where: :$\dbinom n k$ denotes a binomial coefficient :$x^{\overline k}$ denotes $x$ to the $k$ rising.


Free Matroid is Matroid Let $S$ be a finite set. Let $\struct{S, \powerset S}$ be the free matroid of $S$. Then $\struct{S, \powerset S}$ is a matroid.


Real Function with Strictly Negative Derivative is Strictly Decreasing If $\forall x \in \openint a b: \map {f'} x < 0$, then $f$ is strictly decreasing on $\closedint a b$.


Addition Rule for Gaussian Binomial Coefficients/Formulation 2 :$\dbinom n m_q = \dbinom {n - 1} m_q q^m + \dbinom {n - 1} {m - 1}_q$


Open Set of Irreducible Space is Irreducible Let $T = \struct {S, \tau}$ be an irreducible topological space. Let $U$ be a non-empty open set of $T$. Then $U$ is irreducible in its induced subspace topology.


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


Order Isomorphism between Linearly Ordered Spaces is Homeomorphism Let $\struct {S_1, \le_1, \tau_1}$ and $\struct {S_2, \le_2, \tau_2}$ be linearly ordered spaces. Let $\phi: S_1 \to S_2$ be an order isomorphism from $\struct {S_1, \le_1}$ to $\struct {S_2, \le_2}$. Then $\phi$ is a homeomorphism from $\struct {S_1, \tau_1}$ to $\struct {S_2, \tau_2}$.


Non-Zero Real Numbers Closed under Multiplication The set of non-zero real numbers is closed under multiplication: :$\forall x, y \in \R_{\ne 0}: x \times y \in \R_{\ne 0}$


Completely Multiplicative Function of Quotient Let $f: \R \to \R$ be a completely multiplicative function. Then: :$\forall x, y \in \R, y \ne 0: f \left({\dfrac x y}\right) = \dfrac {f \left({x}\right)} {f \left({y}\right)}$ whenever $f \left({y}\right) \ne 0$.


Identity Matrix is Permutation Matrix An identity matrix is an example of a permutation matrix.


Trivial Module is Not Unitary Let $\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\struct {R, +_R, \circ_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be the trivial $R$-module, such that: :$\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$ Then unless $R$ is a ring with unity and $G$ contains only one element, this is ''not'' a unitary module.


Modulo Addition is Closed/Real Numbers Let $z \in \R$ be a real number. Then addition modulo $z$ on the set of residue classes modulo $z$ is closed: :$\forall \eqclass x z, \eqclass y z \in \R_z: \eqclass x z +_z \eqclass y z \in \R_z$.


Open Rational-Number Balls form Neighborhood Basis in Real Number Line Let $\R$ be the real number line with the usual (Euclidean) metric. Let $a \in \R$ be a point in $\R$. Let $\BB_a$ be defined as: :$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \Q_{>0} }$ that is, the set of all open $\epsilon$-balls of $a$ for rational $\epsilon$. Then the $\BB_a$ is a basis for the neighborhood system of $a$.


Numbers which are Sum of Increasing Powers of Digits The following integers are the sum of the increasing powers of their digits taken in order begin as follows: :$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$ {{OEIS|A032799}}


Minimal Element of an Ordinal The minimal element of any nonempty ordinal is the empty set. That is, if $S$ is a nonempty ordinal, $\bigcap S = \O$


Left Operation is Anticommutative The left operation is anticommutative: :$\forall x, y: x \leftarrow y = y \leftarrow x \iff x = y$


Farey Sequence has Convergent Subsequences for all x in Closed Unit Interval Consider the Farey sequence: :$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$ Every element of the closed real interval $\closedint 0 1$ is the limit of a subsequence of $\sequence {a_n}$.


Generating Function of Sequence by Index Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Then: :$z G' \left({z}\right)$ is the generating function for the sequence $\left\langle{n a_n}\right\rangle$ where $G' \left({z}\right)$ is the derivative of $G \left({z}\right)$ {{WRT|Differentiation}} $z$.


Mapping Assigning to Element Its Lower Closure is Isomorphism Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set where :$\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$, :$\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$ Let $P = \left({K\left({I}\right), \precsim'}\right)$ be an ordered subset of $I$ where :$K\left({I}\right)$ denotes the compact subset of $I$. Let $f:S \to K\left({I}\right)$ be a mapping such that :$\forall x \in S: f\left({x}\right) = x^\preceq$ Then $f$ is an order isomorphism between $L$ and $P$.


Bounds for Rank of Subset Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $A \subseteq S$ be subset of $S$. Then: :$0 \le \map \rho A \le \size A$


Primitive of Reciprocal of Hyperbolic Cosine of a x :$\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$


Doubly Sequenced Generating Function for Binomial Coefficients Let $\sequence {a_{m n} }$ be the doubly subscripted sequence defined as: :$\forall m, n \in \N_{\ge 0}: a_{m n} = \dbinom n m$ where $\dbinom n m$ denotes a binomial coefficient. Then the generating function for $\sequence {a_{m n} }$ is given as: :$\map G {w, z} = \dfrac 1 {1 - z - w z}$


Non-Zero Integers under Multiplication are not Subgroup of Reals Let $\struct {\Z_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero integers under multiplication. Let $\struct {\R_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero real numbers under multiplication. Then, while $\struct {\Z_{\ne 0}, \times}$ is closed, it is not a subgroup of $\struct {\R_{\ne 0}, \times}$.


Closed Subset of Real Numbers with Lower Bound contains Infimum Consider the real number line as a metric space under the usual metric. Let $A \subseteq \R$ such that $A$ is closed in $\R$ and $A \neq \varnothing$. Let $A$ be bounded below. Then $A$ contains its infimum.


Empty Set Satisfies Topology Axioms Let $T = \struct {\O, \set \O}$ where $\O$ denotes the empty set. Then $T$ satisfies the open set axioms for a topological space.


Newton-Girard Formulas/Examples/Order 2 :$\displaystyle \sum_{a \mathop \le i \mathop < j \mathop \le b} x_i x_j = \dfrac 1 2 \left({\left({\sum_{i \mathop = a}^b x_i}\right)^2 - \left({\sum_{i \mathop = a}^b {x_i}^2}\right)}\right)$


Floquet's Theorem Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$. Let $\Phi \left({t}\right)$ be a fundamental matrix of the Floquet system $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$. Then $\Phi \left({t + T}\right)$ is also a fundamental matrix. Moreover, there exists: : A nonsingular, continuously differentiable matrix function $\mathbf P \left({t}\right)$ with period $T$ : A constant (possibly complex) matrix $\mathbf B$ such that: ::$\Phi \left({t}\right) = \mathbf P \left({t}\right) e^{\mathbf Bt}$


Pumping Lemma for Regular Languages Let $\mathcal L_3$ be the set of regular languages. {{explain|Is it a set? Does this need to be proved? Intuition would suggest that it would be a class.}} Then the following holds: $\forall L \in \mathcal L_3: \exists n_0 \in \N_0: \forall z \in L: \left|{z}\right| > n_0 \implies \exists u, v, w$ such that: :$z = u \cdot v \cdot w$ :$\left|{v}\right| > 0$ :$\left|{u v}\right| < n_0$ :$\forall i \in \N_0: u \cdot v^i \cdot w \in L$


Gelfond's Constant is Transcendental '''Gelfond's constant''': :$e^\pi$ is transcendental.


Relative Frequency is Probability Measure The relative frequency model is a probability measure.


Free Monoid is Unique Let $S$ be a set. Let $\struct {M, i}$ and $\struct {N, j}$ be free monoids over $S$. Then there is a unique monoid isomorphism $f: M \to N$ such that: :$\size f \circ i = j$ :$\size {f^{-1} } \circ j = i$ where $\size {\, \cdot \,}$ denotes the underlying set functor on $\mathbf{Mon}$.


Graph of Nonlinear Additive Function is Dense in the Plane Let $f: \R \to \R$ be an additive function which is not linear. Then the graph of $f$ is dense in the real number plane.


Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1 Let $\map \Re p > 1$. Then the $p$-series: :$\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$ converges absolutely.


Idempotent Ring has Characteristic Two/Corollary :$\forall x \in R: -x = x$


Rational Numbers with Denominators Coprime to Prime under Addition form Group Let $p$ be a prime number. Let $\Q_p$ denote the set: :$\set {\dfrac r s : s \perp p}$ where $s \perp p$ denotes that $s$ is coprime to $p$. Then $\struct {\Q_p, +}$ is a group.


Countable Particular Point Space is Lindelöf Let $T = \struct {S, \tau_p}$ be a countable particular point space. Then $T$ is a Lindelöf apace.


First Order ODE/exp y dx + (x exp y + 2 y) dy = 0 is an exact differential equation with solution: :$x e^y + y^2 = C$


Floor Function is Replicative The floor function is a replicative function in the sense that: :$\displaystyle \forall n \in \Z_{> 0}: \sum_{k \mathop = 0}^{n - 1} \left \lfloor{x + \frac k n}\right \rfloor = \left \lfloor{n x}\right \rfloor$


Integers with Prime Values of Sigma Function The sequence of integer whose $\sigma$ value is prime begins: {{begin-eqn}} {{eqn | l = \map \sigma 2       | r = 3 }} {{eqn | l = \map \sigma 4       | r = 7 }} {{eqn | l = \map \sigma 6       | r = 13 }} {{eqn | l = \map \sigma {16}       | r = 31 }} {{eqn | l = \map \sigma {25}       | r = 31 }} {{eqn | l = \map \sigma {64}       | r = 127 }} {{eqn | l = \map \sigma {289}       | r = 307 }} {{end-eqn}} {{OEIS|A023194}}


Equivalence of Definitions of Transitive Closure (Set Theory) Let $x$ and $y$ be sets. {{TFAE|def = Transitive Closure (Set Theory)|view = Transitive Closure|context = Set Theory}}


Differential Entropy of Continuous Uniform Distribution Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then the differential entropy of $X$, $\map h X$, is given by:  :$\map h X = \map \ln {b - a}$


Characteristics of Floor and Ceiling Function/Real Domain Let $f: \R \to \Z$ be an integer-valued function which satisfies both of the following: :$(1): \quad \map f {x + 1} = \map f x + 1$ :$(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$ Then it is not necessarily the case that either: :$\forall x \in \R: \map f x = \floor x$ or: :$\forall x \in \R: \map f x = \ceiling x$


Equation of Trochoid Consider a circle $C$ of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on on the line of a radius of $C$ at a distance $b$ from the center of $C$. Let $P$ be on the y-axis when the center of $C$ is also on the y-axis. Consider the trochoid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = a \theta  - b \sin \theta$ :$y = a - b \cos \theta$


Existence of Completely Hausdorff Space which is not Regular There exists at least one example of a topological space which is a completely Hausdorff space, but is not also a regular space.


Prime Number is Deficient Let $p$ be a prime number. Then $p$ is deficient.


Recurrence Relation for Sequence of mth Powers of Fibonacci Numbers Let $m \in \Z_{>0}$ be a (strictly) positive integer. Then: :$\displaystyle \sum_{k \mathop \in \Z} \dbinom m k_\mathcal F \left({-1}\right)^{\left\lceil{\left({m - k}\right) / 2}\right\rceil} {F_{n + k} }^{m - 1} = 0$ where: :$\dbinom m k_\mathcal F$ denotes a Fibonomial coefficient :$F_{n + k}$ denotes the $n + k$th Fibonacci number :$\left\lceil{\, \cdot \,}\right\rceil$ denotes the ceiling function


Sum of Squares of Sum and Difference :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$


Multiplicative Group of Complex Numbers is not Direct Product of Reals with Reals Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers. Then the direct product $\struct {\R_{\ne 0}, \times} \times \struct {\R_{\ne 0}, \times}$ is not isomorphic with $\struct {\C_{\ne 0}, \times}$.


Circumcenter of Triangle is Orthocenter of Medial Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the medial triangle of $\triangle ABC$. Let $K$ be the circumcenter of $\triangle ABC$. Then $K$ is the orthocenter of $\triangle DEF$.


Interior may not equal Exterior of Exterior Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of the underlying set $S$ of $T$. Let $A^e$ be the  exterior of $A$. Let $A^\circ$ be the  interior of $A$. Then it is not necessarily the case that: :$A^{ee} = A^\circ$


Inverses for Rational Addition Each element $x$ of the set of rational numbers $\Q$ has an inverse element $-x$ under the operation of rational number addition: :$\forall x \in \Q: \exists -x \in \Q: x + \paren {-x} = 0 = \paren {-x} + x$


Bottom is Unique Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then $S$ has at most one bottom.


Recurrence Relation for Number of Derangements on Finite Set The number of derangements $D_n$ on a finite set $S$ of cardinality $n$ is: :$D_n = \paren {n - 1} \paren {D_{n - 1} + D_{n - 2} }$ where $D_1 = 0$, $D_2 = 1$.


Elements of Module with Equal Images under Linear Transformations form Submodule Let $G$ and $H$ be $R$-modules. Let $\phi$ and $\psi$ be linear transformations from $G$ into $H$. Then the set $S = \set {x \in G: \map \phi x = \map \psi x}$ is a submodule of $G$.


Sum of Infinite Series of Product of nth Power of cos 2 theta by 2n+1th Multiple of Sine Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty \paren {\cos 2 \theta}^n \sin \paren {2 n + 1} \theta       | r = \sin \theta + \cos 2 \theta \sin 3 \theta + \paren {\cos 2 \theta}^2 \sin 5 \theta + \paren {\cos 2 \theta}^3 \sin 7 \theta + \cdots       | c =  }} {{eqn | r = \dfrac {\csc \theta} 2       | c =  }} {{end-eqn}}


Trisecting the Angle by Compass and Straightedge Construction is Impossible There is no compass and straightedge construction for the trisection of the general angle.


First Subsequence Rule Let $T = \left({A, \tau}\right)$ be a Hausdorff space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $T$. Suppose $\left \langle {x_n} \right \rangle$ has two convergent subsequences with different limit. Then $\left \langle {x_n} \right \rangle$ is divergent.


Trivial Relation is Largest Equivalence Relation The trivial relation $\mathcal T$ on $S$ is the largest equivalence in $S$, in the sense that: :$\forall \mathcal E \subseteq S \times S: \mathcal E \subseteq \mathcal T$ where $\mathcal E$ denotes a general equivalence relation.


Scattered Space is T0 Let $T = \struct {S, \tau}$ be a scattered topological space. Then $T$ is also a $T_0$ (Kolmogorov) space.


Natural Number is not Subset of its Union Let $n \in \N$ be a natural number as defined by the von Neumann construction. Then, except in the degenerate case where $n = 0$, it is not the case that: :$n \subseteq \bigcup n$


Beta Function expressed using Gamma Functions Let $\map \Beta {x, y}$ denote the Beta function. Then: :$\map \Beta {x, y} = \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$ where $\Gamma$ is the Gamma function:


Primitive of Reciprocal of x cubed by Root of a squared minus x squared cubed :$\displaystyle \int \frac {\mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C$


Reciprocal of 99 The decimal expansion of the reciprocal of $99$ is: :$\dfrac 1 {99} = 0 \cdotp \dot 0 \dot 1$ {{OEIS|A000035}}


Real Function with Negative Derivative is Decreasing If $\forall x \in \openint a b: \map {f'} x \le 0$, then $f$ is decreasing on $\closedint a b$.


Linear Second Order ODE/y'' - 3 y' + 2 y = 14 sine 2 x - 18 cosine 2 x The second order ODE: :$(1): \quad y'' - 3 y' + 2 y = 14 \sin 2 x - 18 \cos 2 x$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$


Real Number Line is Non-Meager Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is non-meager.


Leibniz's Rule/Real Valued Functions Let $f, g : \R^n \to \R$ be real valued functions, $k$ times differentiable on some open set $U \subseteq \R^n$. Let $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ be a multiindex indexed by $\set {1, \ldots, n}$ with $\size \alpha \le k$. For $i \in \set {1, \ldots, n}$, let $\partial_i$ denote the partial derivative: :$\partial_i = \dfrac {\partial} {\partial {x_i} }$ Let $\partial^\alpha$ denote the partial differential operator: :$\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \cdots \partial_n^{\alpha_n}$ Then as functions on $U$, we have: :$\displaystyle \map {\partial^\alpha} {f g} = \sum_{\beta \mathop \le \alpha} \binom \alpha \beta \paren {\partial^\beta f} \paren {\partial^{\alpha - \beta} g}$


Number of Permutations with Repetition Set $S$ be a set of $n$ elements. Let $\sequence T_m$ be a sequence of $m$ terms of $S$. Then there are $n^m$ different instances of $\sequence T_m$.


Primitive of Reciprocal of p squared plus square of q by Cosine of a x :$\displaystyle \int \frac {\d x} {p^2 + q^2 \cos^2 a x} = \frac 1 {a p \sqrt{p^2 + q^2} } \arctan \frac {p \tan a x} {\sqrt {p^2 + q^2} } + C$


Bottom-Up Form of Top-Down Grammar defines same Formal Language Let $\mathcal L$ be a formal language. Let $\mathcal T$ be a top-down grammar for $\mathcal L$. Let $\mathcal B$ be the bottom-up form of $\mathcal T$. Then $\mathcal B$ is also a formal grammar for $\mathcal L$.


Lines are Subspaces of Plane The one-dimensional subspaces of $\R^2$ are precisely the homogeneous lines of plane analytic geometry.


Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers The group of Gaussian integer units under complex multiplication: :$\struct {U_\C, \times} = \struct {\set {1, i, -1, -i}, \times}$ forms a subgroup of the multiplicative group of complex numbers.


Lagrange Polynomial Approximation Let $f: D \to \R$ be $n+1$ times differentiable in an interval $I \subset \R$. Let $x_0, \dotsc, x_n \in I$ be pairwise distinct points. Let $P$ be the Lagrange Interpolation Formula of degree at most $n$ such that $P \left({x_i}\right) = f \left({x_i}\right)$ for all $i = 0, \dotsc, n$. Let $R \left({x}\right) = f \left({x}\right) - P \left({x}\right)$. Then, for every $x \in I$, there exists $\xi$ in the interval spanned by $x$ and $x_i$, $i = 0, \dotsc, n$, such that: :$R \left({x}\right) = \dfrac{\left({x - x_0}\right) \left({x - x_1}\right) \dotsm \left({x - x_n}\right) f^{\left({n + 1}\right)} \left({\xi}\right)} {\left({n + 1}\right)!}$


Inequality for Ordinal Exponentiation Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal and let $y > 0$. Let $\sequence {a_i}$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\sequence {b_i}$ be a sequence of natural numbers. Then: :$\displaystyle \paren {\sum_{i \mathop = 1}^n x^{a_i} \times b_i}^y \le x^{a_1 \mathop \times y} \times \paren {b_1 + 1}$


Residue of Quotient Let $f$ and $g$ be functions holomorphic on some region containing $a$.  Let $g$ have a zero of multiplicity $1$ at $a$.  Then: :$\Res {\dfrac f g} a = \dfrac {\map f a} {\map {g'} a}$ {{explain|definition of "Res", by a link to Residue}}


Linear Transformations of Commutative Scalar Ring Let $R$ be a commutative ring. Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules. Let $\mathcal L_R \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$. Then $\mathcal L_R \left({G, H}\right)$ is a submodule of the $R$-module $H^G$. If $H$ is a unitary module, then so is $\mathcal L_R \left({G, H}\right)$.


Relative Homotopy is Equivalence Relation Let $X$ and $Y$ be topological spaces. Let $K \subseteq X$ be a (possibly empty) subset of $X$. Let $\map \CC {X, Y}$ be the set of all continuous mappings from $X$ to $Y$. Define a relation $\sim$ on $\map \CC {X, Y}$ as: :$f \sim g$ {{iff}} $f$ and $g$ are homotopic relative to $K$. Then $\sim$ is an equivalence relation.


Basis for Topology on Cartesian Product Let $T_1 = \struct{A_1, \tau_1}$ and $T_2 = \struct{A_2, \tau_2}$ be topological spaces. Let $A_1 \times A_2$ be the Cartesian product of $A_1$ and $A_2$. Let $\PP = \set{ U_1 \times U_2 : U_1 \in \tau_1, U_2 \in \tau_2}$ Then $\PP$ is a synthetic basis on $A_1 \times A_2$.


Complex Modulus of Quotient of Complex Numbers Let $z_1, z_2 \in \C$ be complex numbers such that $z_2 \ne 0$. Let $\cmod z$ denote the modulus of $z$. Then: :$\cmod {\dfrac {z_1} {z_2} } = \dfrac {\cmod {z_1} } {\cmod {z_2} }$


Equivalence of Definitions of Boundary Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. {{TFAE|def = Boundary (Topology)|view = boundary of $H$}}


Inscribing Regular 15-gon in Circle In a given circle, it is possible to inscribe a regular 15-gon. {{:Euclid:Proposition/IV/16}}


Abelianization of Free Group is Free Abelian Group Let $X$ be a set. Let $\struct {F_X, \iota}$ be a free group on $X$. Let $F_X^{\mathrm {ab} }$ be its abelianization. Let $\pi : F_X \to F_X^{\mathrm {ab} }$ be the quotient group epimorphism. Then $\struct {F_X^{\mathrm {ab} }, \pi \circ \iota}$ is a free abelian group on $X$.


Totally Disconnected but Connected Set must be Singleton Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be both totally disconnected and connected. Then $H$ is a singleton.


Left Identity Element is Idempotent Let $\struct {S, \circ}$ be an algebraic structure. Let $e_L \in S$ be a left identity with respect to $\circ$. Then $e_L$ is idempotent under $\circ$.


Element in Preimage of Image under Mapping Let $f: S \to T$ be a mapping. Then: :$\forall x \in S: x \in f^{-1} \sqbrk {\map f x}$


Locally Euclidean Space is Locally Compact Let $M$ be a locally Euclidean space of some dimension $d$. Then $M$ is locally compact.


Equivalence of Definitions of Algebraically Closed Field Let $K$ be a field.  {{TFAE|def = Algebraically Closed Field}}


Partition of Facets of Rubik's Cube Let $S$ denote the set of the facets of Rukik's cube. Then $S$ can be partitioned as follows: :$S = \set {S_C \mid S_E \mid S_Z}$ where: :$S_C$ denotes the set of corner facets :$S_E$ denotes the set of edge facets :$S_Z$ denotes the set of center facets.


Field Norm of Quaternion is not Norm Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The field norm of $\mathbf x$: :$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$ is not a norm in the abstract algebraic context of a division ring.


Multiplication of Real Numbers is Right Distributive over Subtraction {{:Euclid:Proposition/V/6}} That is, for any number $a$ and for any integers $m, n$: :$m a - n a = \paren {m - n} a$


Strict Ordering on Integers is Transitive Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | lo= \land       | l = \eqclass {c, d} {}       | o = <       | r = \eqclass {e, f} {}       | c =  }} {{eqn | ll= \implies       | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {e, f} {}       | c =  }} {{end-eqn}} That is, strict ordering on the integers is transitive.


Real Power of Strictly Positive Real Number is Strictly Positive Let $x$ be a strictly positive real number. Let $y$ be a real number. Then: :$x^y > 0$ where $x^y$ denotes $x$ raised to the $y$th power.


Conditions for Internal Group Direct Product Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $H_1, H_2 \le G$. Then $G$ is the internal group direct product of $H_1$ and $H_2$ {{iff}}: :$(1): \quad \forall h_1 \in H_1, h_2 \in H_2: h_1 \circ h_2 = h_2 \circ h_1$ :$(2): \quad G = H_1 \circ H_2$ :$(3): \quad H_1 \cap H_2 = \set e$ Condition $(1)$ can also be stated as: :$(1): \quad$ Either $H_1$ or $H_2$ is normal in $G$


Identity Morphism of Product Let $\mathbf C$ be a metacategory. Let $C$ and $D$ be objects of $\mathbf C$, and let $C \times D$ be a binary product for $C$ and $D$. Then: :$\operatorname{id}_{\left({C \mathop \times D}\right)} = \operatorname{id}_C \times \operatorname{id}_D$ where $\operatorname{id}$ denotes an identity morphism, and $\times$ signifies a product of morphisms.


Conditions Satisfied by Linear Code Let $p$ be a prime number. Let $\Z_p$ be the set of residue classes modulo $p$. Let $C := \tuple {n, k}$ be a linear code of a master code $\map V {n, p}$. Then $C$ satisfies the following conditions: :$(C \, 1): \quad \forall \mathbf x, \mathbf y \in C: \mathbf x + \paren {-\mathbf y} \in C$ :$(C \, 2): \quad \forall \mathbf x \in C, m \in \Z_p: m \times \mathbf x \in C$ where $+$ and $\times$ are the operations of codeword addition and codeword multiplication respectively. {{expand|Add a page defining the difference between codewords.}}


Equivalence of Definitions of Non-Invertible Matrix Let $\struct {R, +, \circ}$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\mathcal M_R} n, +, \times}$. The following definitions for $\mathbf A$ to be non-invertible are equivalent:


Power Series Expansion for Logarithm of x/Formulation 2 {{begin-eqn}} {{eqn | l = \ln x        | r = \sum_{n \mathop = 1}^\infty \paren {\frac {x - 1} x}^n }} {{eqn | r = \frac {x - 1} x + \frac 1 2 \paren {\frac {x - 1} x}^2 + \frac 1 3 \paren {\frac {x - 1} x}^3 + \cdots }} {{end-eqn}} valid for all $x \in \R$ such that $x \ge \dfrac 1 2$.


Biconditional in terms of NAND :$p \iff q \dashv \vdash \paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \uparrow \paren {p \uparrow q}$ where $\iff$ denotes logical biconditional and $\uparrow$ denotes logical NAND.


Images of Elements under Repeated Composition of Injection form Equivalence Classes Let $S$ be a set. Let $f: S \to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \ldots, f^n, \ldots$ be defined as: :$\forall n \in \N: \map {f^n} x = \begin {cases} x & : n = 0 \\ \map f x & : n = 1 \\ \map f {\map {f^{n - 1} } x} & : n > 1 \end{cases}$ Let $\mathcal R \subseteq S \times S$ be the relation on $S$ defined as: :$\mathcal R = \set {\tuple {a, b} \in S \times S: \exists k \in \Z: b = \map {f^k} a \lor \exists j \in \Z: a = \map {f^j} b}$ Then $\mathcal R$ is an equivalence relation.


Primitive of Power of x by Arcsine of x over a :$\displaystyle \int x^m \arcsin \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt{a^2 - x^2} } + C$


Equivalence of Definitions of Distance to Nearest Integer Function The following definitions of the distance to nearest integer function $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ are equivalent:


LCM of Three Numbers Let $a, b, c \in \Z: a b c \ne 0$. The lowest common multiple of $a, b, c$, denoted $\lcm \set {a, b, c}$, can always be found. {{:Euclid:Proposition/VII/36}}


Second Order ODE/x y'' - y' = 3 x^2 The second order ODE: :$(1): \quad x y'' - y' = 3 x^2$ has the general solution: :$y = x^3 + \dfrac {C_1 x^2} 2 + C^2$


Real Number Line is Locally Compact Hausdorff Space Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is a locally compact Hausdorff space.


Equivalence of Definitions of Generalized Ordered Space Let $\struct {S, \preceq}$ be a totally ordered set. Let $\tau$ be a topology for $S$. {{TFAE|def = Generalized Ordered Space}}


Multiplicative Identity is Unique Let $\struct {F, +, \times}$ be a field. Then the multiplicative identity $1_F$ of $F$ is unique.


Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form :$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$


Integers which are Sigma for 3 Integers The sequence of integers which are the $\sigma$ value of $3$ different integers begins: :$24, 42, 48, 60, 84, 90, \ldots$ {{OEIS|A007372}}


Primitive of Reciprocal of Cotangent of a x :$\displaystyle \int \frac {\d x} {\cot a x} = \frac {-\ln \size {\cos a x} } a + C$


Shortest Distance between Two Points is Straight Line The shortest distance between $2$ points is a straight line.


Diagonals of Rhombus Intersect at Right Angles Let $ABCD$ be a rhombus. The diagonals $AC$ and $BD$ of $ABCD$ intersect each other at right angles.


Condition for Open Extension Space to be First-Countable Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$. Then $T^*_{\bar p}$ is a first-countable space {{iff}} $T$ is.


Lune of Hippocrates Take the circle whose center is $A$ and whose radius is $AB = AC = AD = AE$. Let $C$ be the center of a circle whose radius is $CD = CF = CE$. :400px Consider the lune $DFEB$. Its area is equal to that of the square $AEGC$.


Primitive of Cube of Cosine of a x :$\displaystyle \int \cos^3 a x \rd x = \frac {\sin a x} a - \frac {\sin^3 a x} {3 a} + C$


Sine of Half Side for Spherical Triangles :$\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$ where $S = \dfrac {A + B + C} 2$.


Equivalence of Definitions of Hereditarily Compact Let $T = \left({S, \tau}\right)$ be a topological space. {{TFAE|def = Hereditarily Compact Space}}


Triplets of Products of Two Distinct Primes The following triplets of consecutive positive integers are the smallest in which each number is the product of $2$ distinct prime numbers: :$33, 34, 35$ :$85, 86, 87$ :$93, 94, 95$ :$141, 142, 143$ :$201, 202, 203$ :$213, 214, 215$ :$217, 218, 219$


Product Category is Category Let $\mathbf C$ and $\mathbf D$ be metacategories. Then the product category $\mathbf C \times \mathbf D$ is a metacategory.


Composition of Ring Monomorphisms is Ring Monomorphism Let: * $\left({R_1, +_1, \circ_1}\right)$ * $\left({R_2, +_2, \circ_2}\right)$ * $\left({R_3, +_3, \circ_3}\right)$ be rings. Let: * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$ be (ring) monomorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) monomorphism.


Set of Positive Integers does not form Ring Let $\Z_{\ge 0}$ denote the set of positive integers. Then the algebraic structure $\struct {\Z_{\ge 0}, +, \times}$ does not form a ring.


Equivalence Classes induced by Derivative Function on Set of Functions Let $X$ be the set of real functions $f: \R \to \R$ which possess continuous derivatives. Let $\mathcal R \subseteq X \times X$ be the equivalence relation on $X$ defined as: :$\mathcal R = \set {\tuple {f, g} \in X \times X: D f = D g}$ where $D f$ denotes the first derivative of $f$. Then the equivalence classes of $\mathcal R$ are defined as: :$\map {\eqclass f {\mathcal R} } x = \set {g \in X: \exists c \in \R: \forall x \in \R: \map g x = \map f x + c}$ That is, it consists of the set of all real functions $f \in X$ which differ by a real constant.


Right Operation is Anticommutative The right operation is anticommutative: :$\forall x, y: x \rightarrow y = y \rightarrow x \iff x = y$


Subring is not necessarily Ideal Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +_S, \circ_S}\right)$ be a subring of $R$. Then it is not necessarily the case that $S$ is also an ideal of $R$.


Coin-Tossing Modeled by Bernoulli Trial The act of coin-tossing can be modeled as a Bernoulli trial. This applies whether the coin is fair or biased.


Open Ray is Open in GO-Space/Definition 2 Let $\struct {S, \preceq, \tau}$ be a generalized ordered space by Definition 2. That is: :Let $\struct {S, \preceq}$ be a totally ordered set. :Let $\struct {S, \tau}$ be a topological space. Let there be: :a linearly ordered space $\struct {S', \preceq', \tau'}$ and: :a mapping $\phi: S \to S'$ which is both: ::a $\preceq$-$\preceq'$ order embedding :and: ::a $\tau$-$\tau'$ topological embedding. Let $p \in S$. Then: :$p^\prec$ and $p^\succ$ are $\tau$-open where: :$p^\prec$ is the strict lower closure of $p$ :$p^\succ$ is the strict upper closure of $p$.


Union of Set of Dense-in-itself Sets is Dense-in-itself Let $T$ be a topological space. Let $\mathcal F \subseteq \mathcal P \left({T}\right)$ such that  :every element of $\mathcal F$ is dense-in-itself. Then the union $\bigcup \mathcal F$ is also dense-in-itself.


Smallest Non-Palindromic Number with Palindromic Square $26$ is the smallest non-palindromic integer whose square is palindromic.


Positive Image of Point of Continuous Real Function implies Positive Closed Interval of Domain Let $f: \R \to \R$ be a continuous real function. Let $a \in \R$ such that $f \left({a}\right) > 0$. Then: :$\exists k \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \left[{a - \delta \,.\,.\, a + \delta}\right]: f \left({x}\right) \ge k$


Equivalence of Definitions of Ordering/Proof 1 The following definitions of ordering are equivalent:


Primitive of x by Inverse Hyperbolic Cosine of x over a :$\displaystyle \int x \cosh^{-1} \frac x a \rd x = \begin{cases} \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \cosh^{-1} \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C & : \cosh^{-1} \dfrac x a > 0 \\ \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \cosh^{-1} \dfrac x a + \dfrac {x \sqrt {x^2 - a^2} } 4 + C & : \cosh^{-1} \dfrac x a < 0 \end{cases}$


Convex Set is Star Convex Set Let $V$ be a vector space over $\R$ or $\C$. Let $A \subseteq V$ be a non-empty convex set. Then $A$ is a star convex set, and every point in $A$ is a star center.


De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice Let $\left({S, \wedge, \vee, \preceq}\right)$ be a uniquely complemented lattice. Then the following are equivalent: $(1):\quad \forall p, q \in S: \neg p \vee \neg q = \neg \left({p \wedge q}\right)$ $(2):\quad \forall p, q \in S: \neg p \wedge \neg q = \neg \left({p \vee q}\right)$ $(3):\quad \forall p, q \in S: p \preceq q \iff \neg q \preceq \neg p$ $(4):\quad \left({S, \wedge, \vee, \preceq}\right)$ is a distributive lattice.


Generating Function for mth Terms of Sequence Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\omega = e^{2 i \pi / m} = \cos \dfrac {2 \pi} m + i \sin \dfrac {2 \pi} m$. Then for $r \in \Z$ such that $0 \le r < m$: :$\displaystyle \sum_{n \bmod m \mathop = r} a_n z^n = \dfrac 1 m \sum_{0 \mathop \le k \mathop < m} \omega^{-k r} G \left({\omega^k z}\right)$


Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x :$\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x} = \frac {-2 \coth 2 a x} a + C$


Laplace Transform Exists if Function Piecewise Continuous and of Exponential Order Let $f$ be a real function which is: :piecewise continuous in every closed interval $\closedint 0 N$ :of exponential order $\gamma$ for $t > N$ Then the Laplace transform $\map F s$ of $\map f t$ exists for all $s > \gamma$.


Zero of Subfield is Zero of Field The zero of $\struct {K, +, \times}$ is also $0$.


Convergent Sequence in P-adic Numbers has Unique Limit Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$. Then $\sequence {x_n}$ can have at most one limit.


T3 1/2 Space is not necessarily T2 Space Let $T = \struct {S, \tau}$ be a be a $T_{3 \frac 1 2}$ space. Then it is not necessarily the case that $T$ is a $T_2$ (Hausdorff) space.


Floor equals Ceiling iff Integer :$\left \lfloor {x}\right \rfloor = \begin{cases} \left \lceil {x}\right \rceil & : x \in \Z \\ \left \lceil {x}\right \rceil - 1 & : x \notin \Z \\ \end{cases}$ or equivalently: :$\left \lceil {x}\right \rceil = \begin{cases} \left \lfloor {x}\right \rfloor & : x \in \Z \\ \left \lfloor {x}\right \rfloor + 1 & : x \notin \Z \\ \end{cases}$ where $\Z$ is the set of integers.


Subspace of Real Functions of Differentiability Class Let $\mathbb J = \set {x \in \R: a < x < b}$ be an open interval of the real number line $\R$. Let $\map {\CC^{\paren m} } {\mathbb J}$ be the set of all continuous real functions on $\mathbb J$ in differentiability class $m$. Then $\struct {\map {\CC^{\paren m} } {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.


Symmetric Difference with Universe :$\mathbb U * S = \map \complement S$ where: :$\mathbb U$ denotes the universe :$*$ denotes symmetric difference.


Sum of Projections/Binary Case Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then $P + Q$ is a projection {{iff}} $\Rng P \perp \Rng Q$.


Sum of Reciprocals of Fourth Powers Alternating in Sign {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^4}       | r = \frac 1 {1^4} - \frac 1 {2^4} + \frac 1 {3^4} - \frac 1 {4^4} + \cdots       | c =  }} {{eqn | r = \dfrac {7 \pi^4} {720}       | c =  }} {{end-eqn}}


Sign of Cotangent Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \cot x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ }} {{eqn | l = \cot x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ }} {{end-eqn}} where $\cot$ is the real cotangent function.


Complete Graph is Hamiltonian for Order Greater than 2 Let $n \in \Z$ be an integer such that $n > 2$. Let $K_n$ denote the complete graph of order $n$. Then $K_n$ is Hamiltonian.


Bounded Class is Set Let $B$ be a class. Let it be assumed that $B$ is a subclass of a basic universe $V$. Let $B$ be bounded by a set $x$. Then $B$ is itself a set.


Limit Points in Uncountable Fort Space Let $T = \struct {S, \tau_p}$ be an uncountable Fort space. Let $U \subseteq S$ be a countably infinite subset of $S$. Then $p$ is the only limit point of $U$.


P-Sequence Space with P-Norm forms Normed Vector Space $P$-Sequence Space with $p$-norm forms normed vector space.


Identity of Submonoid is not necessarily Identity of Monoid Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$. Let $\struct {T, \circ}$ be a submonoid of $\struct {S, \circ}$ whose identity is $e_T$. Then it is not necessarily the case that $e_T = e_S$.


Modulo Operation/Examples/1.1 mod 1 :$1 \cdotp 1 \bmod 1 = 0 \cdotp 1$


Sequence of Implications of Connectedness Properties Let $P_1$ and $P_2$ be connectedness properties and let: :$P_1 \implies P_2$ mean: :If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$. Then the following sequence of implications holds: {| |- | align="center" | || | align="center" | || | align="center" | Ultraconnected || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Arc-Connected || | align="center" | $\implies$ || | align="center" | Path-Connected || |- | align="center" | || | align="center" | || | align="center" | $\Big\Downarrow$ || |- | align="center" | Irreducible || | align="center" | $\implies$ || | align="center" | Connected || |- |}


Hilbert Cube is Compact Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is a compact space.


No Group has Two Order 2 Elements A group can not contain exactly two elements of order $2$.


Existence of Dedekind Completion Let $\left({S, \preceq}\right)$ be an ordered set. Then there exists a Dedekind completion of $S$. That is, there exists a Dedekind complete ordered set $\tilde S$ and an order embedding $\phi: S \to \tilde S$ such that: :For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \to X$, there exists an order embedding $\tilde f: \tilde S \to X$ such that: ::$\tilde f \circ \phi = f$


Angle of Spherical Triangle from Sides 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. Then: :$\cos A = \cosec b \cosec c \paren {\cos a - \cos b \cos c}$


Primitive of x cubed by Root of a squared minus x squared cubed :$\displaystyle \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$


Group is Abelian iff Opposite Group is Itself Let $\left({G, \circ}\right)$ be a  group. Let $\left({G, *}\right)$ be the opposite group to $({G, \circ})$. $\left({G, \circ}\right)$ is an Abelian group {{iff}}: :$\left({G, \circ}\right) = \left({G, *}\right)$


Sierpiński Space is Ultraconnected Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. Then $T$ is ultraconnected.


Open Ball in Real Number Plane under Chebyshev Distance Let $\R^2$ be the real number plane. Let $d_\infty: \R^2 \times \R^2 \to \R$ be the Chebyshev Distance on $\R^2$: : $\displaystyle d_\infty \left({x, y}\right):= \max \left\{ {\left\vert{x_1 - y_1}\right\vert, \left\vert{x_2 - y_2}\right\vert}\right\}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$. For $a \in \R^2$, let $B_\epsilon \left({a}\right)$ be the open $\epsilon$-ball at $a$. Then $B_\epsilon \left({a}\right)$ is the interior of the square centered at $a$ and whose sides are of length $2 \epsilon$ parallel to the coordinate axes.


Steiner-Lehmus Theorem Let $ABC$ be a triangle. Denote the lengths of the angle bisectors through the vertices $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$. Let $\omega_\alpha = \omega_\beta$. Then $ABC$ is an isosceles triangle.


Gaussian Integers form Subgroup of Complex Numbers under Addition The set of Gaussian integers $\Z \sqbrk i$, under the operation of complex addition, forms a subgroup of the set of additive group of complex numbers $\struct {\C, +}$.


Polynomials in Integers is not Principal Ideal Domain Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$. Then $\Z \sqbrk X$ is not a principal ideal domain.


Intersection of Orthocomplements is Orthocomplement of Closed Linear Span Let $H$ be a Hilbert space. Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of closed linear subspaces of $H$. Then: :$\ds \bigcap_{i \mathop \in I} M_i^\perp = \paren {\vee \set {M_i : i \in I} }^\perp$ where: :$\perp$ denotes orthocomplementation :$\vee$ denotes closed linear span.


One-Digit Number is Harshad Let $n$ be a $1$-digit positive integer. Then $n$ is a harshad number.


Concave Real Function is Continuous Let $f$ be a real function which is concave on the open interval $\left({a \,.\,.\, b}\right)$. Then $f$ is continuous on $\left({a \,.\,.\, b}\right)$.


Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $T = \struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \cup \closedint 2 3$ where $\closedint 2 3$ is a closed interval of $\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is a limit point of $B$ in $\R$.


Half-Life of Radioactive Substance Let a radioactive element $S$ decay with a rate constant $k$. Then its half-life $T$ is given by: :$T = \dfrac {\ln 2} k$ :400px


Second Order ODE/(x^2 + 2 y') y'' + 2 x y' = 0 The second order ODE: :$(1): \quad \paren {x^2 + 2 y'} y'' + 2 x y' = 0$ subject to the initial conditions: :$y = 1$ and $y' = 0$ when $x = 0$ has the particular solution: :$y = 1$ or: :$3 y + x^3 = 3$


Left Module Does Not Necessarily Induce Right Module over Ring Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ' : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ'}$ is not necessarily a right module over $\struct {R, +_R, \times_R}$


Pointwise Multiplication on Integer-Valued Functions is Associative Let $f, g, h: S \to \Z$ be integer-valued functions. Let $f \times g: S \to \Z$ denote the pointwise product of $f$ and $g$. Then: :$\paren {f \times g} \times h = f \times \paren {g \times h}$


Non-Palindromes in Base 2 by Reverse-and-Add Process Let the number $22$ be expressed in binary: $10110_2$. When the reverse-and-add process is performed on it repeatedly, it never becomes a palindromic number.


Elements of Abelian Group whose Order Divides n is Subgroup Let $G$ be an abelian group whose identity element is $e$. Let $n \in \Z_{>0}$ be a (strictly) positive integer . Let $G_n$ be the subset of $G$ defined as: :$G_n = \set {x \in G: \order x \divides n}$ where: :$\order x$ denotes the order of $x$ :$\divides$ denotes divisibility. Then $G_n$ is a subgroup of $G$.


Implicit Function Theorem for Lipschitz Contraction at Point Let $M$ and $N$ be metric spaces. Let $M$ be complete. Let $f : M \times N \to M$ be a uniform contraction. Then for all $t\in N$ there exists a unique $g(t) \in M$ such that $f(g(t), t) = g(t)$, and if $f$ is Lipschitz continuous at a point $(g(t),t)$, then $g$ is Lipschitz continuous at $t$.


Factorial Divisible by Binary Root Let $n \in \Z: n \ge 1$. Let $n$ be expressed in binary notation: :$n = 2^{e_1} + 2^{e_2} + \cdots + 2^{e_r}$ where $e_1 > e_2 > \cdots > e_r \ge 0$. Let $n!$ be the factorial of $n$. Then $n!$ is divisible by $2^{n-r}$, but not by $2^{n-r+1}$.


Correctness of Definition of Increasing Mappings Satisfying Inclusion in Lower Closure Let $R = \left({S, \preceq}\right)$ be an ordered set. Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$ Let  :$M = \left({F, \preccurlyeq}\right)$ where :$F = \left\{ {f: S \to {\it Ids}\left({R}\right): f}\right.$ is increasing mapping $\left.{\land \forall x \in S: f\left({x}\right) \subseteq x^\preceq}\right\}$ and :$\preccurlyeq$ is ordering on mappings generated by $\precsim$ where $x^\preceq$ denotes the lower closure of $x$. Then :$M$ is an ordered set.


Vinogradov's Theorem/Minor Arcs For any $B > 0$: :$\displaystyle \int_\mathcal M \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$


Cartesian Plane Rotated with respect to Another Let $\mathbf r$ be a position vector embedded in a Cartesian plane $\CC$ with origin $O$. Let $\CC$ be rotated anticlockwise through an angle $\varphi$ about the axis of rotation $O$. Let $\CC'$ denote the Cartesian plane in its new position. Let $\mathbf r$ be kept fixed during this rotation. Let $\tuple {x, y}$ denote the components of $\mathbf r$ with respect to $\CC$. Let $\tuple {x', y'}$ denote the components of $\mathbf r$ with respect to $\CC'$. Then: {{begin-eqn}} {{eqn | l = x'       | r = x \cos \varphi + y \sin \varphi }} {{eqn | l = y'       | r = -x \sin \varphi + y \cos \varphi }} {{end-eqn}}


Primitive of Error Function :$\displaystyle \int \map \erf x \rd x = x \, \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C$


Pseudocompact Normal Space is Countably Compact Let $T = \struct {S, \tau}$ be a normal space. Then $T$ is pseudocompact {{iff}} $T$ is countably compact.


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


Primitive of Root of a x + b over Power of p x + q :$\displaystyle \int \frac {\sqrt{a x + b} } {\left({p x + q}\right)^n} \ \mathrm d x = \frac {- \sqrt{a x + b} } {\left({n - 1}\right) p \left({p x + q}\right)^{n-1} } + \frac a {2 \left({n - 1}\right) p} \int \frac {\mathrm d x} {\left({p x + q}\right)^{n-1} \sqrt{a x + b} }$


Primitive of x over Cosine of a x {{begin-eqn}} {{eqn | l = \int \frac {x \rd x} {\cos a x}       | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C       | c =  }} {{eqn | r = \dfrac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 + \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} + \cdots} + C       | c =  }} {{end-eqn}}


Summation to n of kth Harmonic Number over k+1 :$\displaystyle \sum_{k \mathop = 1}^n \dfrac {H_k} {k + 1} = \dfrac { {H_{n + 1} }^2 - H_{n + 1}^{\left({2}\right)} } 2$ where: :$H_n$ denotes the $n$th harmonic number :$H_n^{\left({2}\right)}$ denotes a general harmonic number.


Principle of Finite Induction/Zero-Based Let $S \subseteq \N$ be a subset of the natural numbers. Suppose that: :$(1): \quad 0 \in S$ :$(2): \quad \forall n \in \N : n \in S \implies n + 1 \in S$ Then: :$S = \N$


Symmetry Group of Rectangle is Klein Four-Group The symmetry group of the rectangle is the Klein $4$-group.


Root of Equation e^x (x - 1) = e^-x (x + 1) The equation: :$e^x \paren {x - 1} = e^{-x} \paren {x + 1}$ has a root: :$x = 1 \cdotp 19966 \, 78640 \, 25773 \, 4 \ldots$


Bijection between Power Set of nth Initial Section and Initial Section of nth Power of 2 Let $\N_n$ be used to denote the first $n$ non-zero natural numbers: :$\N_n = \set {1, 2, \ldots, n}$ Then there exists a bijection between the power set of $\N_n$ and $\N_{2^n}$.


Fundamental Theorem of Contour Integration Let $D \subseteq \C$ be an open set. Let $f: D \to \C$ be a continuous function. Suppose that $F: D \to \C$ is an antiderivative of $f$. Let $\gamma: \closedint a b \to D$ be a contour in $D$. Then the contour integral: :$\displaystyle \int_\gamma \map f z \rd z =  \map F {\map \gamma b} - \map F {\map \gamma a}$


Stabilizer in Group of Transformations Let $X$ be any set with $n$ elements (where $n \in \Z_{>0}$). Consider the symmetric group on $n$ letters $S_n$ as a group of transformations on $X$. Let $x \in X$. Then the stabilizer of $x$ is isomorphic to $S_{n - 1}$.


Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x :$\displaystyle \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x = \frac \pi {2 b} \tanh \frac {a \pi} {2 b}$


Geometric Mean of two Positive Real Numbers is Between them Let $a, b \in \R$ be real numbers such that $0 < a < b$. Let $\map G {a, b}$ denote the geometric mean of $a$ and $b$. Then: :$a < \map G {a, b} < b$


Closed Subsets of Symmetry Group of Square Recall the symmetry group of the square:


Primitive Root is Generator of Reduced Residue System Let $a$ be a primitive root of $n$. Then: :$\left\{{a, a^2, a^3, \ldots, a^{\phi \left({n}\right)}}\right\}$ where $\phi \left({n}\right)$ is the Euler phi function of $n$, is a reduced residue system of $n$. Thus the first $\phi \left({n}\right)$ powers of $a$ "generates" $R$. We say that $a$ is a '''generator''' of $R$.


Set is Subset of Power Set of Union Let $x$ be a set of sets. Let $\displaystyle \bigcup x$ denote the union of $x$. Let $\powerset {\displaystyle \bigcup x}$ denote the power set of $\displaystyle \bigcup x$. Then: :$x \subseteq \powerset {\displaystyle \bigcup x}$


Absolute Value of Divergent Infinite Product The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$ {{iff}}  $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ diverges to $0$.


Equivalence of Definitions of Secant of Angle Let $\theta$ be an angle. {{TFAE|def = Secant of Angle|view = secant}}


Zero of Field is Unique Let $\struct {F, +, \times}$ be a field. The zero of $F$ is unique.


Second-Countability is not Continuous Invariant Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping. If $T_A$ is a second-countable space, then it does not necessarily follow that $T_B$ is also second-countable.


Absolute Value of Complex Cross Product is Commutative Let $z_1$ and $z_2$ be complex numbers. Let $z_1 \times z_2$ denote the (complex) cross product of $z_1$ and $z_2$. Then: :$\size {z_1 \times z_2} = \size {z_2 \times z_1}$ where $\size {\, \cdot \,}$ denotes the absolute value function.


G-Module is Irreducible iff no Non-Trivial Proper Submodules Let $\left({G, \circ}\right)$ be a finite group. Let $\left({V, \phi}\right)$ be a $G$-module. Then $V$ is an irreducible $G$-module {{iff}} $V$ has no non-trivial proper $G$-submodules.


Either-Or Topology is Non-Meager Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is non-meager.


Relative Complement of Cartesian Product Let $A$ and $B$ be sets. Let $X \subseteq A$ and $Y \subseteq B$. Then: :$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$


Inequality of Hölder Means Let $p, q \in \R_{\ne 0}$ be non-zero real numbers with $p < q$. Let $x_1, x_2, \ldots, x_n \ge 0$ be real numbers. If $p < 0$, then we require that $x_1, x_2, \ldots, x_n > 0$. Then the Hölder mean satisfies the inequality: : $M_p \left({x_1, x_2, \ldots, x_n}\right) \le M_q \left({x_1, x_2, \ldots, x_n}\right)$ Equality holds if and only if $x_1 = x_2 = \cdots = x_n$.


Primitive of Fourth Power of Sine of a x :$\displaystyle \int \sin^4 a x \ \mathrm d x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$


Distance between Linear Codewords is Distance Function Let $\map V {n, p}$ be a master code. Let $d: V \times V \to \Z$ be the mapping defined as: :$\forall u, v \in V: \map d {u, v} =$ the distance between $u$ and $v$ that is, the number of corresponding terms at which $u$ and $v$ are different. Then $d$ defines a distance function in the sense of a metric space.


Condition for Cartesian Product Equivalent to Associated Cardinal Number Let $S$ and $T$ be nonempty sets. Let $\left|{S}\right|$ denote the cardinal number of $S$. Then: :$S \times T \sim \left|{S \times T}\right| \iff S \sim \left|{S}\right| \land T \sim \left|{T}\right|$ where $S \times T$ denotes the cartesian product of $S$ and $T$.


Matroid Contains No Loops iff Empty Set is Flat Let $M = \struct{S, \mathscr I}$ be a matroid. Then: :$M$ contains no loops {{iff}} the empty set is flat.


Integers whose Phi times Tau equal Sigma The positive integers whose Euler $\phi$ function multiplied by its  $\tau$ function equals its $\sigma$ function are: :$1, 3, 14, 42$ {{OEIS|A104905}}


Number of Compositions A $k$-composition of a positive integer $n$ is an ordered $k$-tuple: $c = \left({c_1, c_2, \ldots, c_k}\right)$ such that $c_1 + c_2 + \cdots + c_k = n$ and $c_i $ are strictly positive integers. The number of $k$-compositions of $n$ is $\displaystyle \binom{n-1}{k-1}$ and the total number of compositions of $n$ is $2^{n-1}$ (i.e. for $k = 1, 2, 3, \ldots, n$).