Area between Radii and Whorls of Archimedean Spiral Let $S$ be the Archimedean spiral defined by the equation: :$r = a \theta$ Let $\theta = \theta_1$ and $\theta = \theta_2$ be the two rays from the pole at angles $\theta_1$ and $\theta_b$ to the polar axis respectively. Let $R$ be the figure enclosed by: :$\theta_1$ and $\theta_2$ :the $n$th turn of $S$ and the $n+1$th turn of $S$ The area $\mathcal A$ of $R$ is given by: :$\mathcal A = a^2 \pi \left({\theta_2 - \theta_1}\right) \left({\theta_2 + \theta_1 + 2 \pi \left({2 n + 1}\right)}\right)$ {{Proofread|My algebra may have failed me}}


Cycle does not Contain Subcycles Let $G$ be a cycle graph. Then the only cycle graph that is a subgraph of $G$ is $G$ itself.


Central Limit Theorem Let $X_1, X_2, \ldots$ be a sequence of independent identically distributed random variables with: :Mean $E \left[{X_i}\right] = \mu \in \left({-\infty \,.\,.\, \infty}\right)$ :Variance $V \left({X_i}\right) = \sigma^2 > 0$ Let: : $\displaystyle S_n = \sum_{i \mathop = 1}^n X_i$ Then: :$\displaystyle \frac {S_n - n \mu} {\sqrt {n \sigma^2} } \xrightarrow {D} N \left({0, 1}\right)$ as $n \to \infty$ that is, converges in distribution to a standard normal.


Proportion of Power Let $x$ and $y$ be proportional. {{explain|Establish what types of object $x$ and $y$ are. As it stands here, they could be anything.}} Let $n \in \Z$.  Then $x^n \propto y^n$.


Lifting The Exponent Lemma for Sums for p=2 Let $x, y \in \Z$ be integers with $x + y \ne 0$. Let $n \ge 1$ be an odd natural number. Let: :$2 \divides x + y$ where $\divides$ denotes divisibility. Then: :$\map {\nu_2} {x^n + y^n} = \map {\nu_2} {x + y}$ where $\nu_2$ denotes $2$-adic valuation. </onlyinclude>


Primitive of x by Arccosine of x over a :$\displaystyle \int x \arccos \frac x a \ \mathrm d x = \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arccos \frac x a - \frac {x \sqrt {a^2 - x^2} } 4 + C$


Coproduct on Disjoint Union Let $S_1$ and $S_2$ be sets. Let $S_1 \sqcup S_2 := \paren {S_1 \times \set 1} \cup \paren {S_2 \times \set 2}$ be the disjoint union of $S_1$ and $S_2$. Let $i_1: S_1 \to S_1 \sqcup S_2$ and $i_2: S_2 \to S_1 \sqcup S_2$ be the mappings defined as: :$\forall s_1 \in S_1: \map {i_1} {s_1} = \tuple {s_1, 1}$ :$\forall s_2 \in S_2: \map {i_2} {s_2} = \tuple {s_2, 2}$ Then $\struct {S_1 \sqcup S_2, i_1, i_2}$ is a coproduct of $S_1$ and $S_2$.


Sequence of Palindromic Lucky Numbers The The sequence of lucky numbers which are also palindromic begins: :$1, 3, 7, 9, 33, 99, 111, 141, 151, 171, \ldots$ {{OEIS|A031161}}


Cotangent of Complex Number/Formulation 3 :$\cot \paren {a + b i} = \dfrac {- \cot a - \cot a \coth^2 b} {\cot^2 a + \coth^2 b} + \dfrac {\cot^2 a \coth b - \coth b} {\cot^2 a + \coth^2 b} i$


Order of Element in Group equals its Order in Subgroup Let $G$ be a group. Let $H\ le G$, where $\le$ denotes the property of being a subgroup. Let $x \in H$. Then the order of $x$ in $H$ equals the order of $x$ in $G$.


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


Union of Elements of Power Set Let $S$ be a set. Then: :$\displaystyle S = \bigcup_{X \mathop \in \powerset S} X$ where $\powerset S$ denotes the power set of $S$.


Exponential on Real Numbers is Group Isomorphism Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\R_{> 0}, \times}$ be the multiplicative group of positive real numbers. Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be the mapping: :$x \mapsto \map \exp x$ where $\exp$ is the exponential function. Then $\exp$ is a group isomorphism.


Real Part of Complex Exponential Function Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\exp z$ denote the complex exponential function. Then: :$\map \Re {\exp z} = e^x \cos y$ where: :$\Re z$ denotes the real part of a complex number $z$ :$e^x$ denotes the real exponential function of $x$ :$\cos y$ denotes the real cosine function of $y$.


Primitive of Power of x by Cosine of a x/Corollary :$\displaystyle \int x^m \cos a x \rd x = \sum_{k \mathop = 1}^{m + 1} \paren {m^{\underline {k - 1} } \frac {x^{m + 1 - k} } {a^k} \map {\sin} {x + \dfrac {\pi} 2 \paren {k - 1} } }$ where $m^{\underline {k - 1} }$ denotes the $k - 1$th falling factorial of $m$.


Trace of Sum of Matrices is Sum of Traces Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$. let $\mathbf A + \mathbf B$ debote the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. Then: :$\map \tr {\mathbf A + \mathbf B} = \map \tr {\mathbf A} + \map \tr {\mathbf B}$ where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$.


Existence of Unique Subsemigroup Generated by Subset Let $\struct {S, \circ}$ be a semigroup. Let $\O \subset X \subseteq S$. Let $\struct {T, \circ}$ be the subsemigroup generated by $X$. Then $T = \gen X$ exists and is unique.


Power Reduction Formulas/Hyperbolic Sine Cubed :$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$


Probability Measure on Single-Subset Event Space Let $\EE$ be an experiment whose sample space is $\Omega$. Let $\O \subsetneqq A \subsetneqq \Omega$. Let $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ be the event space of $\EE$. Let $\Pr: \Sigma \to \R$ be a probability measure on $\struct {\Omega, \Sigma}$. Then $\Pr$ has the form: {{begin-eqn}} {{eqn | n = Pr 1       | l = \map \Pr \O       | r = 0 }} {{eqn | n = Pr 2       | l = \map \Pr A       | r = p }} {{eqn | n = Pr 3       | l = \map \Pr {\Omega \setminus A}       | r = 1 - p }} {{eqn | n = Pr 4       | l = \map \Pr \Omega       | r = 1 }} {{end-eqn}} for some $p \in \R$ satisfying $0 \le p \le 1$.


Countable Product of First-Countable Spaces is First-Countable Let $I$ be an indexing set with countable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be first-countable. Then $\struct {S, \tau}$ is also first-countable.


Lexicographic Order on Products of Well-Ordered Sets Let $S$ be a set which is well-ordered by $\preceq$. Let $\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$. Then: :$(1): \quad$ For a given $n \in \N: n > 0$, $\preccurlyeq$ is a well-ordering on the set $T_n$ of all ordered $n$-tuples of $S$; :$(2): \quad \preccurlyeq$ is '''not''' a well-ordering on the set $T$ of ''all'' ordered tuples of $S$.


1 is Limit Point of Sequence in Sierpiński Space Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. The sequence in $T$: :$\sigma = \sequence {0, 1, 0, 1, \ldots}$ has $1$ as a limit point.


Intersection of Plane with Sphere is Circle The intersection of a plane with a sphere is a circle.


Subgroup is Normal iff it contains Product of Inverses A subgroup $H$ of a group $G$ is normal {{iff}}: :$\forall a, b \in G: a b \in H \implies a^{-1} b^{-1} \in H$


5th Cyclotomic Ring is not a Unique Factorization Domain Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. Then $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ is not a unique factorization domain. The following elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are irreducible: :$2$ :$3$ :$1 + i \sqrt 5$ :$1 - i \sqrt 5$


Definite Integral to Infinity of x over Exponential of x minus One :$\displaystyle \int_0^\infty \frac x {e^x - 1} \rd x = \frac {\pi^2} 6$


Transitive Closure Always Exists (Relation Theory) Let $\RR$ be a relation on a set $S$. Then the transitive closure $\RR^+$ of $\RR$ always exists.


Content of Monic Polynomial If $f$ is monic, then $\cont f = \dfrac 1 n$ for some integer $n$.


Real Number Line is First-Countable Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is first-countable.


Sum of Geometric Sequence/Examples/Index to Minus 2 Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$. Then the formula for Sum of Geometric Sequence: :$\displaystyle \sum_{j \mathop = 0}^n x^j = \frac {x^{n + 1} - 1} {x - 1}$ breaks down when $n = -2$: :$\displaystyle \sum_{j \mathop = 0}^{-2} x^j \ne \frac {x^{-1} - 1} {x - 1}$


Paracompact Countably Compact Space is Compact Let $T = \struct {S, \tau}$ be a countably compact space which is also paracompact. Then $T$ is compact.


Primitive of Reciprocal of Square of 1 minus Cosine of a x :$\displaystyle \int \frac {\mathrm d x} {\left({1 - \cos a x}\right)^2} = \frac {-1} {2a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C$


Product is Left Identity Therefore Left Cancellable Let $\struct {S, \circ}$ be a semigroup. Let $e_L \in S$  be a left identity of $S$. Let $a \in S$ such that: :$\exists b \in S: b \circ a = e_L$ Then $a$ is left cancellable in $\struct {S, \circ}$.


Transitivity of Algebraic Extensions Let $E / F / K$ be a tower of field extensions. Let $E$ be algebraic over $F$. Let $F$ be algebraic over $K$. Then $E$ is algebraic over $K$.


Primitive of Reciprocal of x by x fourth minus a fourth :$\displaystyle \int \frac {\mathrm d x} {x \left({x^4 - a^4}\right)} = \frac 1 {4 a^4} {\ln \left\vert{\frac {x^4 - a^4} {x^4} }\right\vert} + C$


Error Detection Capability of Linear Code Let $C$ be a linear code. Let $C$ have a minimum distance $d$. Then $C$ detects $d - 1$ or fewer transmission errors.


Linear Second Order ODE/y'' + 3 y' - 10 y = 6 exp 4 x The second order ODE: :$(1): \quad y'' + 3 y' - 10 y = 6 e^{4 x}$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-5 x} + \dfrac {e^{4 x} } 3$


Arens-Fort Topology is Topology Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $\tau$ is a topology on $T$.


Upper Bounds for Prime Numbers/Result 3 : $\forall n \in \N_{>1}: p \left({n}\right) < 2^n$


Sums of Sequences of Consecutive Squares which are Square The sums of the following sequences of successive squares are themselves square: {{begin-eqn}} {{eqn | l = \sum_{i \mathop = 7}^{29} k^2       | r = 7^2 + 8^2 + \cdots + 29^2       | c =  }} {{eqn | l = \sum_{i \mathop = 7}^{39} k^2       | r = 7^2 + 8^2 + \cdots + 39^2       | c =  }} {{eqn | l = \sum_{i \mathop = 7}^{56} k^2       | r = 7^2 + 8^2 + \cdots + 56^2       | c =  }} {{eqn | l = \sum_{i \mathop = 7}^{190} k^2       | r = 7^2 + 8^2 + \cdots + 190^2       | c =  }} {{end-eqn}}


Cook-Levin Theorem The Boolean Satisfiability Problem is NP-Complete.


Imaginary Part of Sine of Complex Number Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\sin z$ denote the complex sine function. Then: :$\Im \paren {\sin z} = \cos x \sinh y$ where: :$\Im z$ denotes the imaginary part of a complex number $z$ :$\sin$ denotes the complex sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function.


Primitive of Tangent of a x/Secant Form :$\ds \int \tan a x \rd x = \frac {\ln \size {\sec a x} } a + C$


Cartesian Product of Preimage with Image of Relation is Correspondence Let $\mathcal R \subseteq S \times T$ be a relation. Then the restriction of $\mathcal R$ to $\Preimg {\mathcal R} \times \Img {\mathcal R}$ is a correspondence.


Primitive of x by Inverse Hyperbolic Cotangent of x over a :$\displaystyle \int x \coth^{-1} \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \coth^{-1} \frac x a + C$


Alaoglu's Theorem The closed unit ball of the dual of a normed space is compact with respect to the weak* topology. {{rewrite|Rephrase to match usual presentation}}


Rule of Exportation/Formulation 1/Proof 2 :$\paren {p \land q} \implies r \dashv \vdash p \implies \paren {q \implies r}$


Definite Integral from 0 to 2 Pi of Logarithm of a plus b Cosine x :$\displaystyle \int_0^{2 \pi} \map \ln {a + b \cos x} \rd x = 2 \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$


Implication is Left Distributive over Disjunction/Formulation 1/Forward Implication :$p \implies \paren {q \lor r} \vdash \paren {p \implies q} \lor  \paren {p \implies r}$


Newton's Formula for Pi $\pi$ (pi) can be approximated using the formula: :$\pi = \dfrac {3 \sqrt 3} 4 + 24 \paren {\dfrac 2 {3 \times 2^3} - \dfrac 1 {5 \times 2^5} - \dfrac 1 {28 \times 2^7} - \dfrac 1 {72 \times 2^9} - \dfrac 5 {704 \times 2^{11} } - \dfrac 7 {1664 \times 2^{13} } - \cdots}$


Distinct Elements are Parallel iff Pair forms Circuit Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x, y \in S : x \ne y$. Then: :$x$ and $y$ are parallel {{iff}} $\set {x, y}$ is a circuit


Rational Numbers under Multiplication form Commutative Monoid The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a countably infinite commutative monoid.


Field of Quotients of Ring of Polynomial Forms on Reals that yields Complex Numbers Let $\struct {\R, +, \times}$ denote the field of real numbers. Let $X$ be transcendental over $\R$. Let $\R \sqbrk X$ be the ring of polynomials in $X$ over $F$. Consider the field of quotients: :$\R \sqbrk X / \ideal p$ where: :$p = X^2 + 1$ :$\ideal p$ denotes the ideal generated by $p$. Then $\R \sqbrk X / \ideal p$ is the field of complex numbers.


Boundary of Compact Set in Hausdorff Space is Compact Let $X$ be a Hausdorff topological space. Let $K\subset X$ be a compact subspace of $X$. Then its boundary $\partial K$ is compact.


Area under Arc of Cycloid Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ Then the area under one arc of the cycloid is $3 \pi a^2$. That is, the area under one arc of the cycloid is three times the area of the generating circle.


Compact Subset of Compact Space is not necessarily Closed A compact subset of a compact space not necessarily closed.


Divisor of Sum of Coprime Integers Let $a, b, c \in \Z_{>0}$ such that: :$a \perp b$ and $c \divides \paren {a + b}$. where: :$a \perp b$ denotes $a$ and $b$ are coprime :$c \divides \paren {a + b}$ denotes that $c$ is a divisor of $a + b$. Then $a \perp c$ and $b \perp c$. That is, a divisor of the sum of two coprime integers is coprime to both.


Subgroup of Subgroup with Prime Index/Corollary Let $\struct {G, \circ}$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K \subsetneq H$. Let: :$\index G K = p$ where: :$p$ denotes a prime number :$\index G K$ denotes the index of $K$ in $G$. Then: :$H = G$


Sum of Sequence of n by 2 to the Power of n :$\displaystyle \sum_{j \mathop = 0}^n j \, 2^j = n 2^{n + 2} - \paren {n + 1} 2^{n + 1} + 2$


Difference between Kaprekar Number and Square is Multiple of Repunit Let $n$ be a Kaprekar number of $m$ digits. Then: : $n^2 - n = k R_m$ where: :$R_m$ is the $m$-digit repunit :$k$ is an integer.


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


Combination Theorem for Cauchy Sequences/Quotient Rule Suppose $\sequence {y_n}$ does not converge to $0$. Then: :$\exists K \in \N: \forall n > K : y_n \ne 0$ and the sequences: :$\sequence { {x_{K + n} } \paren {y_{K + n} }^{-1} }_{n \mathop \in \N}$ and $\sequence {\paren {y_{K + n} }^{-1} {x_{K + n} } }_{n \mathop \in \N}$ are well-defined and Cauchy sequences.


Inverse not always Unique for Non-Associative Operation Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\circ$ be a non-associative operation. Then for any $x \in S$, it is possible for $x$ to have more than one inverse element.


Biconditional with Contradiction :$p \iff \bot \dashv \vdash \neg p$


Symmetry Group of Line Segment is Group The symmetry group of the line segment is a group.


Set of Upper Closures of Compact Elements is Basis implies Complete Scott Topological Lattice is Algebraic Let $L = \left({S, \preceq, \tau}\right)$ be a complete Scott Definition:Topological Lattice. Let $\mathcal B = \left\{ {x^\succeq: x \in K\left({L}\right)}\right\}$ be a basis of $L$ where :$x^\succeq$ denotes the upper closure of $x$, :$K\left({L}\right)$ denotes the compact subset of $L$. Then $L$ is algebraic.


Crossbar Theorem Let $\triangle ABC$ be a triangle. Let $D$ be a point in the interior of $\triangle ABC$. Then there exists a point $E$ such that $E$ lies on both $AD$ and $BC$.


Derivative of Tangent of Function :$\map {\dfrac \d {\d x} } {\tan u} = \sec^2 u \dfrac {\d u} {\d x}$


Non-Zero Rational Numbers Closed under Multiplication The set of non-zero rational numbers is closed under multiplication.


Primitive of Reciprocal of x squared by x cubed plus a cubed :$\displaystyle \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {6 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } - \frac 1 {a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


False Statement implies Every Statement/Formulation 1/Proof 1 :$\neg p \vdash p \implies q$


Nilpotent Ring Element plus Unity is Unit Let $A$ be a ring with unity. Let $1 \in A$ be its unity. Let $a \in A$ be nilpotent. Then $1 + a$ is a unit of $A$.


Power Series Expansion for Square of Reciprocal of 1-z {{begin-eqn}} {{eqn | l = \dfrac 1 {\paren {1 - z}^2}       | r = \sum_{n \mathop = 0}^\infty \paren {n + 1} z^n }} {{eqn | r = 1 + 2 z + 3 z^2 + 4 z^3 + \cdots }} {{end-eqn}}


Condition for Denesting of Square Root Let $a, b \in \Q_{\ge 0}$. Suppose $\sqrt b \notin \Q$. Then: :$\exists p, q \in \Q: \sqrt {a + \sqrt b} = \sqrt p + \sqrt q$ {{iff}}: :$\exists n \in \Q: a^2 - b = n^2$.


Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Let $U \in \tau_p$ be a countably infinite open set of $T$. Let the elements of $U$ be arranged into a sequence $\sequence {a_i}$ of distinct terms in $T$. Then while every element $x$ of $U$ such that $x \ne p$ is a limit point of $U$, there exists no $x \in U$ such that $x$ is an accumulation point of $\sequence {a_i}$.


Metric Induced by a Pseudometric Let $X$ be a set on which there is a pseudometric $d: X \times X \to \R$. For any $x, y \in X$, let $x \sim y$ denote that $d \left({x, y}\right) = 0$. Let $\left[\!\left[{x}\right]\!\right]$ denote the equivalence class of $x$ under $\sim$. Let $X^*$ be the quotient of $X$ by $\sim$. Then the mapping $d^*: X^* \times X^* \to \R$ defined by: :$d^* \left({\left[\!\left[{x}\right]\!\right], \left[\!\left[{y}\right]\!\right]}\right) = d \left({x, y}\right)$ is a metric. Hence $\left({X^*, d^*}\right)$ is a metric space.


Fibonacci Numbers which equal the Square of their Index The only Fibonacci numbers which equal the square of their index are: {{begin-eqn}} {{eqn | l = F_0       | r = 0 }} {{eqn | l = F_1       | r = 1 }} {{eqn | l = F_{12}       | r = 12^2 = 144 }} {{end-eqn}}


Current in Electric Circuit/L, R in Series/Sinusoidal EMF 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: :$E = E_0 \sin \omega t$ The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} {\sqrt {R^2 - L^2 \omega^2} } \sin \left({\omega t - \alpha}\right) + \left({I_0 - \dfrac {E_0 L \omega} {R^2 + L^2 \omega^2} }\right) e^{-R t / L}$ where $\tan \alpha = \dfrac {L \omega} R$.


Like Unit Vectors are Equal Let $\mathbf a$ and $\mathbf b$ be like vector quantities. Then: :$\mathbf {\hat a} = \mathbf {\hat b}$ where $\mathbf {\hat a}$ and $\mathbf {\hat b}$ denote the unit vectors in the direction of $\mathbf a$ and $\mathbf b$.


Totally Bounded Metric Space is Second-Countable Let $M = \struct {A, d}$ be a metric space which is totally bounded. Then $M$ is second-countable.


Obtuse Triangle Divided into Acute Triangles Let $T$ be an obtuse triangle. Let $T$ be dissected into $n$ acute triangles. Then $n \ge 7$.


Conjunction with Negative Equivalent to Negation of Implication/Formulation 2/Reverse Implication :$\vdash \left({\neg \left({p \implies q}\right)}\right) \implies \left({p \land \neg q}\right)$


Cantor Space is Non-Meager in Itself Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is non-meager in itself.


Sum of Harmonic Numbers approaches Harmonic Number of Product of Indices Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let: : $T \left({m, n}\right) := H_m + H_n - H_{m n}$ where $H_n$ denotes the $n$th harmonic number. Then as $m$ or $n$ increases, $T \left({m, n}\right)$ never increases, and reaches its minimum when $m$ and $n$ approach infinity.


Quadrature of Parabola Let $T$ be a parabola. Consider the parabolic segment bounded by an arbitrary chord $AB$. Let $C$ be the point on $T$ where the tangent to $T$ is parallel to $AB$. Let Then the area $S$ of the parabolic segment $ABC$ of $T$ is given by: :$S = \dfrac 4 3 \triangle ABC$


Class Equality is Symmetric Let $A$ and $B$ be classes. Let $=$ denote class equality. Then: :$A = B \implies B = A$


Euler's Equation/Independent of y' Let $y$ be a mapping. Let $J$ a functional be such that :$\ds J \sqbrk y = \int_a^b \map F {x,y} \rd x$ Then the corresponding Euler's Equation can be reduced to: :$F_y = 0$  Furthermore, this is an algebraic equation.


Absolute Value of Convergent Infinite Product Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge to $a \in \mathbb K$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm {a_n}$  converges to $\norm{a}$.


First-Countability is Hereditary Let $T = \struct {S, \tau}$ be a topological space which is first-countable. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then $T_H$ is first-countable.


Boundedness of Metric Space by Open Ball Let $M = \struct {X, d}$ be a metric space. Let $M' = \struct {Y, d_Y}$ be a subspace of $M$. Then $M'$ is bounded in $M$ {{iff}}: :$\exists x \in M, \epsilon \in \R_{>0}: Y \subseteq \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$. Simply put: a subspace is bounded {{iff}} it can be fitted inside an open ball.


Uncountable Fort Space is not First-Countable Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a first-countable space.


Definite Integral of Fourier Series at Ends of Interval Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$. Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$. Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$. Consider the real function: :$\map F x = \displaystyle \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$ Then: :$\map F \pi = \map F {-\pi} = \dfrac {a_0 \pi} 2$


Pluperfect Digital Invariant has less than 61 Digits Let $n \in \Z_{>0}$ be a pluperfect digital invariant. Then $n$ has less than $61$ digits.


Rouché's Theorem Let $f$ and $g$ be complex-valued functions which are holomorphic in the interior of some simply connected region $D$. Let $\left\vert{g \left({z}\right)}\right\vert < \left\vert{f \left({z}\right)}\right\vert$ on the boundary of $D$. Then $f$ and $f + g$ have the same number of zeroes in the interior of $D$ counted up to multiplicity.


Superset of Neighborhood in Topological Space is Neighborhood Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $N$ be a neighborhood of $x$ in $T$. Let $N \subseteq N' \subseteq S$. Then $N'$ is a neighborhood of $x$ in $T$.


Squeeze Theorem for Filter Bases Let $\struct {S, \le, \tau}$ be a linearly ordered space. Let $F_1$, $F_2$, and $F_3$ be filter bases in $S$. Let: :$\forall T \in F_1: \exists M \in F_2: \forall x \in M: \exists y \in T: y \le x$ That is: :for each $T \in F_1$, $F_2$ has an element $M$ such that all elements of $M$ succeed some element of $T$. Similarly, let: :$\forall U \in F_3: \exists N \in F_2: \forall x \in N: \exists y \in U: x \le y$ That is: :for each $U \in F_3$, $F_2$ has an element $N$ such that all elements of $N$ precede some element of $U$. Let $F_1$ and $F_3$ each converge to a point $p \in S$. Then $F_2$ converges to $p$.


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


Consecutive Primes of form 4n+1 The sequence of $16$ consecutive prime numbers beginning from $207 \, 622 \, 273$ are all of the form $4 n + 1$.


Local Connectedness is not Preserved under Continuous Mapping Let $\struct {A, \tau_1}$ and $\struct {B, \tau_2}$ be topological spaces. Let $f: A \to B$ be a continuous mapping. Let $\struct {\Img f, \tau_3}$ be the image of $f$ with the subspace topology of $B$. Let $\struct {A, \tau_1}$ be locally connected. Then it is not necessarily the case that $\struct {\Img f, \tau_3}$ is also locally connected.


Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\Z, +}$ be the additive group of integers. Let $\R / \Z$ denote the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$. Let $x + \Z$ denote the coset of $\Z$ by $x \in \R$. Then $x + \Z$ is of finite order {{iff}} $x$ is rational.


Probability Generating Function of One :$\Pi_X \left({1}\right) = 1$


Primitive of Reciprocal of Cosine of a x/Logarithm of Secant plus Tangent Form :$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\sec a x + \tan a z}$


Discrete Uniform Distribution gives rise to Probability Measure Let $\EE$ be an experiment. Let the probability space $\struct {\Omega, \Sigma, \Pr}$ be defined as: :$\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ :$\Sigma = \powerset \Omega$ :$\forall A \in \Sigma: \map \Pr A = \dfrac 1 n \card A$ where: :$\powerset \Omega$ denotes the power set of $\Omega$ :$\card A$ denotes the cardinality of $A$. Then $\Pr$ is a probability measure on $\struct {\Omega, \Sigma}$.


Norm of Unit of Normed Division Algebra Let $\struct {A_F, \oplus}$ be a normed division algebra. Let the unit of $\struct {A_F, \oplus}$ be $1_A$. Then: :$\norm {1_A} = 1$ where $\norm {1_A}$ denotes the norm of $1_A$.


Alexandroff Extension of Rational Number Space is Sequentially Compact Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a sequentially compact space.


Summation over Finite Set is Well-Defined Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S$ be a finite set. Let $f: S \to \mathbb A$ be a mapping. Let $n$ be the cardinality of $S$. let $\N_{<n}$ be an initial segment of the natural numbers. Let $g, h: \N_{<n} \to S$ be bijections. Then we have an equality of indexed summations of the compositions $f \circ g$ and $f \circ h$: :$\displaystyle\sum_{i \mathop = 0}^{n - 1} \map f {\map g i} = \sum_{i \mathop = 0}^{n - 1} \map f {\map h i}$ That is, the definition of summation over a finite set does not depend on the choice of the bijection $g: S \to \N_{< n}$.


Complement of Symmetric Relation Let $\mathcal R \subseteq S \times S$ be a relation. Then $\mathcal R$ is symmetric {{iff}} its complement $\relcomp {S \times S} {\mathcal R} \subseteq S \times S$ is also symmetric.


Diffuse Measure of Countable Set Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Suppose that for all $x \in X$, the singleton $\left\{{x}\right\}$ is in $\Sigma$. Suppose further that $\mu$ is a diffuse measure. Let $E \in \Sigma$ be a countable measurable set. Then $\mu \left({E}\right) = 0$.


Supremum of Ideals is Upper Adjoint implies Lattice is Continuous Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below up-complete lattice. Let $\mathit{Ids}\left({L}\right)$ be the set of all ideals in $L$. Let $P = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an ordered set where $\mathord \precsim = \subseteq\restriction_{\mathit{Ids}\left({L}\right)\times \mathit{Ids}\left({L}\right)}$ Let $f: \mathit{Ids}\left({L}\right) \to S$ be a mapping such that :$\forall I \in \mathit{Ids}\left({L}\right): f\left({I}\right) = \sup I$ Let $f$ be an upper adjoint of a Galois connection. Then $L$ is continuous.


Orbit of Subgroup under Coset Action is Coset Space Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ be the power set of $G$. Let $H \in \powerset G$ be a subgroup of $G$. Let $*$ be the group action on $H$ defined as: :$\forall g \in G: g * H = g \circ H$ where $g \circ H$ is the (left) coset of $g$ by $H$. Then the orbit of $H$ in $\powerset G$ is the (left) coset space of $H$: :$\Orb H = G / H^l$


Primitive of x by Exponential of a x by Cosine of b x :$\displaystyle \int x e^{a x} \cos b x \ \mathrm d x = \frac {x e^{a x} \left({a \cos b x + b \sin bx}\right)} {a^2 + b^2} - \frac {e^{a x} \left({\left({a^2 - b^2}\right) \cos b x - 2 a b \sin bx}\right)} {\left({a^2 + b^2}\right)^2} + C$


Three-Way Exclusive Or and Equivalence Let $p \iff q$ be the biconditional operator, and $p \oplus q$ be the exclusive or operator. Then: : $p \iff q \iff r \dashv \vdash p \oplus q \oplus r$


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


Prime between n and 9 n divided by 8 Let $n \in \Z$ be an integer such that $n \ge 48$. Then there exists a prime number $p$ such that $n < p < \dfrac {9 n} 8$.


Scalar Multiplication by Zero gives Zero Vector Let $\mathbf a$ be a vector quantity. Let $0 \mathbf a$ denote the scalar product of $\mathbf a$ with $0$. Then: :$0 \mathbf a = \bszero$ where $\bszero$ denotes the zero vector.


Affine Group of One Dimension is Group Let $\map {\operatorname {Af}_1} \R$ be the $1$-dimensional affine group on $\R$. Then $\map {\operatorname {Af}_1} \R$ is a group.


Opposite Group is Group Let $\struct {G, \circ}$ be a group. Let $\struct {G, *}$ be the opposite group to $G$. Then $\struct {G, *}$ is a group.


Chapman-Kolmogorov Equation Let $X$ be a discrete state-space Markov chain with $n$-step transition probability matrix: :$\mathbf P^{\paren n} = \sqbrk {\map {p^{\paren n} } {j, k} }_{j, k \mathop \in S}$ where: :$\map {p^{\paren n} } {j, k} = \condprob {X_{m + n} = k} {X_m = j} = {p_{j k} }^{\paren n}$ is the $n$-step transition probability. Then: :$\mathbf P^{\paren {n + m} } = \mathbf P^{\paren n} \mathbf P^{\paren m}$ or equivalently: :$\displaystyle {p_{i j} }^{\paren {n + m} } = \sum_{k \mathop \in S} {p_{i k} }^{\paren n} {p_{k j} }^{\paren m}$


Higher-Aleph Complement Topology is Topology Let $T = \left({S, \tau}\right)$ be an $\aleph_n$ complement space. Then $\tau$ is a topology on $T$.


Inverse Secant of Imaginary Number :$\sec^{-1} x = \pm i \sech^{-1} x$


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


Set of All Mappings of Cartesian Product Let $R$, $S$, and $T$ be sets. Then: :$R^{S \times T} \sim \left({ R^S }\right)^T$


Separation Properties in Open Extension of Particular Point Topology Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton. Let $T^*_{\bar q} = \struct {S^*_q, \tau^*_{\bar q} }$ be an open extension space of $T$. Then: :$T^*_{\bar q}$ is a $T_0$ (Kolmogorov) space. :$T^*_{\bar q}$ is a $T_4$ (space. :$T^*_{\bar q}$ is not a $T_1$ (Fréchet) space. :$T^*_{\bar q}$ is not a $T_5$ (space.


Unordered Pairs Exist Let $A$ and $B$ be classes, which may be either sets or proper classes. Then: :$\forall A, B: \left\{{A, B}\right\} \in U$ where $U$ is the universal class.


Subgroup Generated by Infinite Order Element is Infinite Let $G$ be a group. Let $a \in G$ be of infinite order. Let $\gen a$ be the subgroup generated  by $a$. Then $\gen a$ is of infinite order.


Mellin Transform of Heaviside Step Function/Corollary :$\mathcal M \left\{ {u\left({c - t}\right)}\right\} \left({s}\right) = \dfrac {c^s} s$ for $c > 0, \Re \left({s}\right) > 0$


Rectangles with Equal Bases and Equal Altitudes are Congruent Let $ABCD$ and $EFGH$ be rectangles. Then $ABCD$ and $EFGH$ are congruent if: : the base of $ABCD$ equals the base of $EFGH$ : the altitude of $ABCD$ equals the altitude of $EFGH$.


Non-Trivial Particular Point Topology is not T3 Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton. Then $T$ is not a $T_3$ space.


Commutativity of Parameters of Beta Function :$\map \Beta {x, y} = \map \Beta {y, x}$


Triangular Numbers in Geometric Sequence The numbers: :$1, 6, 36$ are the smallest triangular numbers in geometric sequence.


Existence of Unique Subgroup Generated by Subset Let $\struct {G, \circ}$ be a group. Let $\O \subset S \subseteq G$. Let $\struct {H, \circ}$ be the subgroup generated by $S$. Then $H = \gen S$ exists and is unique. Also, $\struct {H, \circ}$ is the intersection of all of the subgroups of $G$ which contain the set $S$: :$\displaystyle \gen S = \bigcap_i {H_i}: S \subseteq H_i \le G$


Convergent Sequence in Set of Integers Let $\left \langle {x_n}\right \rangle_{n \in \N}$ be a sequence in the set $\Z$ of integers considered as a subspace of the real number line $\R$ under the Euclidean metric. Then $\left \langle {x_n}\right \rangle_{n \in \N}$ converges in $\R$ to a limit {{iff}}: :$\exists k \in \N: \forall m \in \N: m > k: x_m = x_k$ That is, {{iff}} the sequence reaches some value of $\Z$ and "stays there".


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


Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space of $3$ dimensions: Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Then $\left\lvert{\mathbf a \times \mathbf b}\right\rvert$ equals the area of the parallelogram two of whose sides are $\mathbf a$ and $\mathbf b$.


Edgeless Graphs of Order n are Isomorphic Let $n \in \Z_{>0}$ be a positive integer. Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be edgeless graphs of order $n$. Then $G_1$ and $G_2$ are isomorphic.


Fourth Power is Sum of 2 Triangular Numbers Let $n \in \Z$ be an integer. Then: :$\exists a, b \in \Z_{\ge 0}: n^4 = T_a + T_b$ where $T_a$ and $T_b$ are triangular numbers. That is, the $4$th power of an integer equals the sum of two triangular numbers.


Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers Let $\alpha, \beta \in \C$ be complex numbers. Then: :$\left\lvert{\alpha + \beta}\right\rvert^2 + \left\lvert{\alpha - \beta}\right\rvert^2 = 2 \left\lvert{\alpha}\right\rvert^2 + 2 \left\lvert{\beta}\right\rvert^2$


Commutativity of Incidence Matrix with its Transpose for Symmetric Design Let $A$ be the incidence matrix of a symmetric design. Then: : $A A^\intercal = A^\intercal A$ where $A^\intercal$ is the transpose of $A$.


Primitive of Product of Cosecant and Cotangent :$\ds \int \csc x \cot x \rd x = -\csc x + C$ where $C$ is an arbitrary constant.


Vector Space with Standard Affine Structure is Affine Space Let $E$ be a vector space. Let $\left({\mathcal E, E, +, -}\right)$ be the standard affine structure on $E$. Then with this structure, $\mathcal E$ is an affine space.


219 Fedorov Groups There are $219$ Fedorov groups, if chiral copies are considered the same.


Equivalence of Definitions of Division Ring {{TFAE|def = Division Ring}} A '''division ring''' is a ring with unity $\struct {R, +, \circ}$ with the following properties:


Group is Hausdorff iff has Closed Discrete Subgroup A topological group is Hausdorff {{iff}} it has a closed discrete subgroup.


Image of Set Difference under Mapping/Corollary 1 Let $f: S \to T$ be a mapping. Let $S_1 \subseteq S_2 \subseteq S$. Then: :$\relcomp {f \sqbrk {S_2} } {f \sqbrk {S_1} } \subseteq f \sqbrk {\relcomp {S_2} {S_1} }$ where $\complement$ (in this context) denotes relative complement.


Measure Invariant on Generator is Invariant Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space. Let $\theta: X \to X$ be an $\Sigma / \Sigma$-measurable mapping. Suppose that $\Sigma$ is generated by $\mathcal G \subseteq \mathcal P \left({X}\right)$. Also, let $\mathcal G$ satisfy the following: :$(1):\quad \forall G, H \in \mathcal G: G \cap H \in \mathcal G$ :$(2):\quad$ There exists an exhausting sequence $\left({G_n}\right)_{n \in \N} \uparrow X$ in $\mathcal G$ such that: :::$\quad \forall n \in \N: \mu \left({G_n}\right) < +\infty$ Suppose furthermore that, for all $G \in \mathcal G$, $\mu$ satisfies: :$(3):\quad \mu \left({\theta^{-1} \left({G}\right) }\right) = \mu \left({G}\right)$ Then $\mu$ is a $\theta$-invariant measure.


Generator of Vector Space is Basis iff Cardinality equals Dimension :$G$ is a basis for $E$ {{iff}} $\card G = n$.


Hilbert Sequence Space is not Sigma-Compact Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is not $\sigma$-compact.


Primitive of Reciprocal of x squared by x cubed plus a cubed squared :$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^3 + a^3}\right)^2} = \frac {-1} {a^6 x} - \frac {x^2} {3 a^6 \left({x^3 + a^3}\right)} - \frac 4 {3 a^6} \int \frac {x \ \mathrm d x} {x^3 + a^3}$


Integral Operator is Linear/Corollary 2 :$\forall \alpha \in \R: \map T {\alpha f} = \alpha \map T f$


Completely Irreducible implies Meet Irreducible Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $p \in S$. Then if $p$ is completely irreducible, then $p$ is meet irreducible.


Primitive of x by Hyperbolic Secant of a x :$\displaystyle \int x \sech a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C$


Simple Graph whose Vertices Incident to All Edges Let $G = \struct {V, E}$ be a simple graph whose vertices are incident to all its edges. Then $G$ is either: :the star graph $S_2$, which is also the complete graph $K_2$ :an edgeless graph of any order.


Representation of Integers in Golden Mean Number System The positive integers $n$ are represented in the golden mean number system in their simplest form $S_n$ as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="left" style = "padding: 2px 10px" | $S_n$  |- | align="right" style = "padding: 2px 10px" | $1$ | align="left" style = "padding: 2px 10px" | $1$ |- | align="right" style = "padding: 2px 10px" | $2$ | align="left" style = "padding: 2px 10px" | $10 \cdotp 01$ |- | align="right" style = "padding: 2px 10px" | $3$ | align="left" style = "padding: 2px 10px" | $100 \cdotp 01$ |- | align="right" style = "padding: 2px 10px" | $4$ | align="left" style = "padding: 2px 10px" | $101 \cdotp 01$ |- | align="right" style = "padding: 2px 10px" | $5$ | align="left" style = "padding: 2px 10px" | $1000 \cdotp 1001$ |- | align="right" style = "padding: 2px 10px" | $6$ | align="left" style = "padding: 2px 10px" | $1010 \cdotp 0001$ |- | align="right" style = "padding: 2px 10px" | $7$ | align="left" style = "padding: 2px 10px" | $10000 \cdotp 0001$ |- | align="right" style = "padding: 2px 10px" | $8$ | align="left" style = "padding: 2px 10px" | $10001 \cdotp 0001$ |- | align="right" style = "padding: 2px 10px" | $9$ | align="left" style = "padding: 2px 10px" | $10010 \cdotp 0101$ |- | align="right" style = "padding: 2px 10px" | $10$ | align="left" style = "padding: 2px 10px" | $10100 \cdotp 0101$ |- | align="right" style = "padding: 2px 10px" | $11$ | align="left" style = "padding: 2px 10px" | $10101 \cdotp 0101$ |- | align="right" style = "padding: 2px 10px" | $12$ | align="left" style = "padding: 2px 10px" | $100000 \cdotp 101001$ |- | align="right" style = "padding: 2px 10px" | $13$ | align="left" style = "padding: 2px 10px" | $100010 \cdotp 001001$ |- | align="right" style = "padding: 2px 10px" | $14$ | align="left" style = "padding: 2px 10px" | $100100 \cdotp 001001$ |- | align="right" style = "padding: 2px 10px" | $15$ | align="left" style = "padding: 2px 10px" | $100101 \cdotp 001001$ |- | align="right" style = "padding: 2px 10px" | $16$ | align="left" style = "padding: 2px 10px" | $101000 \cdotp 100001$ |}


Primitive of x by Square of Tangent of a x :$\displaystyle \int x \tan^2 a x \ \mathrm d x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \left\vert{\cos a x}\right\vert - \frac {x^2} 2 + C$


Probability Generating Function of Shifted Random Variable Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$. Let $k \in \Z_{\ge 0}$ be a positive integer. Let $Y$ be a discrete random variable such that $Y = X + m$. Then :$\Pi_Y \left({s}\right) = s^m \Pi_X \left({s}\right)$. where $\Pi_Y \left({s}\right)$ is the probability generating function of $Y$.


Rule of Material Implication/Formulation 2/Reverse Implication : $\vdash \left({\neg p \lor q}\right) \implies \left({p \implies q}\right)$


Homeomorphism may Exist between Non-Comparable Topologies Let $S$ be a set. Let $T_1 = \struct {S, \tau_1}$ and $T_2 = \struct {S, \tau_2}$ be topological spaces defined on the underlying set $S$. Let $\tau_1$ and $\tau_2$ be non-comparable. Then it may possibly be the case that $T_1$ and $T_2$ are homeomorphic.


Equivalence of Definitions of Connected Topological Space {{TFAE|def = Connected Topological Space}} Let $T = \struct {S, \tau}$ be a topological space.


Double Negation Elimination implies Law of Excluded Middle Let the Law of Double Negation Elimination be supposed to hold: :$\neg \neg p \vdash p$ Then the Law of Excluded Middle likewise holds: :$\vdash p \lor \neg p$


Multiindices under Addition form Commutative Monoid Let $Z$ be the set of multiindices. Let $+$ denote the addition of multiindices. Then $\left({Z, +}\right)$ is a commutative monoid.


Four Color Theorem for Finite Maps implies Four Color Theorem for Infinite Maps Suppose that any finite planar graph can be assigned a proper vertex $k$-coloring such that $k \le 4$. Then the same is true of any infinite planar graph.


Partial Derivatives of tan^2 (x^2 - y^2) Let: :$\map f {x, y} = \map {\tan^2} {x^2 - y^2}$ Then: {{begin-eqn}} {{eqn | l = \map {f_1} {x, y}       | r = 4 x \map \tan {x^2 - y^2} \map {\sec^2} {x^2 - y^2} }} {{eqn | l = \map {f_2} {1, 2}       | r = 8 \tan 3 \sec^2 3 }} {{end-eqn}}


Rational Number Space is not Scattered Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is not scattered.


Definite Integral from 0 to Pi of Cosine of m x by Cosine of n x Let $m, n \in \Z$ be integers. Then: :$\displaystyle \int_0^\pi \cos m x \cos n x \rd x = \begin{cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end{cases}$ That is: :$\displaystyle \int_0^\pi \cos m x \cos n x \rd x = \dfrac \pi 2 \delta_{m n}$ where $\delta_{m n}$ is the Kronecker delta.


Ring Epimorphism Preserves Unity Let $A$ be a ring with unity $1$. Let $B$ be a ring. Let $f: A \to B$ be a ring epimorphism. Then $\map f 1$ is a unity of $B$.


Floor Function/Examples/Floor of One Half :$\floor {\dfrac 1 2} = 0$


Primitive of x over Power of x squared plus a squared :$\displaystyle \int \frac {x \ \mathrm d x} {\left({x^2 + a^2}\right)^n} = \frac {-1} {2 \left({n - 1}\right) \left({x^2 + a^2}\right)^{n - 1} }$


Generating Function for Sequence of Sum over k to n of Reciprocal of k by n-k Let $\sequence {a_n}$ be the sequence whose terms are defined as: :$\forall n \in \Z_{\ge 0}: a_n = \displaystyle \sum_{k \mathop = 1}^{n - 1} \dfrac 1 {k \paren {n - k} }$ Then $\sequence {a_n}$ has the generating function $\map G z$ such that: :$\map G z = \paren {\ln \dfrac 1 {1 - z} }^2$ and whose terms are: :$a_n = \dfrac {2 H_{n - 1} } n$


Exterior of Finite Union equals Intersection of Exteriors Let $T$ be a topological space. Let $n \in \N$. Let $\forall i \in \closedint 1 n: H_i \subseteq T$. Then: :$\displaystyle \paren {\bigcup_{i \mathop = 1}^n H_i}^e = \bigcap_{i \mathop = 1}^n H_i^e$ where $H_i^e$ denotes the exterior of $H_i$.


Fibonacci Number of Even Index by Golden Mean Modulo 1 Let $n \in \Z$ be an integer. Then: :$F_{2 n} \phi \bmod 1 = 1 - \phi^{-2 n}$ :$F_n$ denotes the $n$th Fibonacci number :$\phi$ is the golden mean: $\phi = \dfrac {1 + \sqrt 5} 2$


Overflow Theorem Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.


Equivalence of Definitions of Homeomorphic Topological Spaces Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces. Let $f: T_\alpha \to T_\beta$ be a bijection. {{TFAE|def = Homeomorphism (Topological Spaces)|view = homeomorphism}}


Exponential Sequence is Eventually Strictly Positive Let $\sequence {E_n}$ be the sequence of real functions $E_n: \R \to \R$ defined as: :$\map {E_n} x = \paren {1 + \dfrac x n}^n$ Then, for each $x \in \R$ and for sufficiently large $n \in \N$, $\map {E_n} x$ is positive. That is: :$\forall x \in \R: \forall n \in \N: n \ge \ceiling {\size x} \implies \map {E_n} x > 0$ where $\ceiling x$ denotes the ceiling of $x$.


Integral Representation of Dirichlet Beta Function in terms of Gamma Function :$\displaystyle \map \beta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 + e^{-2 x} } \rd x$


Way Below Compact is Topological Compact Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$ Let $x \in \tau$. Then :$x$ is compact in $L$ (it means: $x \ll x$) {{iff}} :$T_x$ is compact (topologically) where $T_x = \left({x, \tau_x}\right)$ denotes the topological subspace of $x$.


Cardinality of Mapping Let $S$ be a finite set whose cardinality is $n$: :$\card S = n$ Let $T$ be a non-empty set Let $f: S \to T$ be a mapping. Then: :$\card f = n$


Orthogonal Trajectories/Cardioids Consider the one-parameter family of curves of cardioids given in polar form as: :$(1): \quad r = c \paren {1 + \cos \theta}$ Its family of orthogonal trajectories is given by the equation: :$r = c \paren {1 - \cos \theta}$ :600px


Circumscribing Regular Pentagon about Circle About a given circle, it is possible to circumscribe a regular pentagon. {{:Euclid:Proposition/IV/12}}


Distance Between Points of Same Latitude along Parallel of Latitude Let $J$ and $K$ be points on Earth's surface that have the same latitude. Let $JK$ be the length of the arc joining $JK$ measured along the parallel of latitude on which they both lie. Let $R$ denote the center of the parallel of latitude holding $J$ and $K$. Let $\operatorname {Long}_J$ and $\operatorname {Long}_K$ denote the longitude of $J$ and $K$ respectively, measured in degrees. Let $\operatorname {Lat}_J$ denote the latitude of $J$ (and $K$). Then:  :$JK \approx 60 \times \size {\operatorname {Long}_J - \operatorname {Long}_K} \cos \operatorname {Lat}_J$ nautical miles


Irrational Number Space is not Weakly Sigma-Locally Compact Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is not weakly $\sigma$-locally compact.


PGF of Sum of Random Number of Discrete Random Variables Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let: :$N, X_1, X_2, \ldots$ be independent discrete random variables such that the $X$'s have the same probability distribution. Let: * $\Pi_N \left({s}\right)$ be the PGF of $N$ * $\Pi_X \left({s}\right)$ be the PGF of each of the $X$'s. Let: :$Z = X_1 + X_2 + \ldots + X_N$ Then: :$\Pi_Z \left({s}\right) = \Pi_N \left({\Pi_X \left({s}\right)}\right)$


Successive Solutions of Phi of n equals Phi of n + 2 $7$ and $8$ are two successive integers which are solutions to the equation: :$\map \phi n = \map \phi {n + 2}$


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


Cosine of Integer Multiple of Argument/Formulation 1 {{begin-eqn}} {{eqn | l = \cos n \theta       | r = \dfrac 1 2 \paren {\paren {2 \cos \theta }^n - \dfrac n 1 \paren {2 \cos \theta }^{n - 2} + \dfrac n 2 \dbinom {n - 3} 1 \paren {2 \cos \theta }^{n - 4} - \dfrac n 3 \dbinom {n - 4} 2 \paren {2 \cos \theta }^{n - 6} + \cdots }       | c =  }} {{eqn | r = \dfrac 1 2 \paren {\paren {2 \cos \theta }^n + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac n k \dbinom {n - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{n - 2 k } }       | c =  }} {{end-eqn}}


Greatest Common Divisor divides Lowest Common Multiple Let $a, b \in \Z$ such that $a b \ne 0$. Then: :$\gcd \set {a, b} \divides \lcm \set {a, b}$ where: :$\lcm$ denotes lowest common multiple :$\gcd$ denotes greatest common divisor. :$\divides$ denotes divisibility.


Ordinal Multiplication via Cantor Normal Form/Infinite Exponent Let $x$ and $y$ be ordinals. Let $x > 1$. Let $y \ge \omega$ where $\omega$ denotes the minimal infinite successor set. Let $\left\langle{a_i}\right\rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\left\langle{b_i}\right\rangle$ be a sequence of ordinals such that $0 < b_i < x$ for all $1 \le i \le n$. Then: :$\displaystyle \sum_{i \mathop = 1}^n \left({x^{a_i} b_i}\right) \times x^y = x^{a_1 \mathop + y}$


Sum of Ring Products is Subring of Commutative Ring Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $\left({S, +, \circ}\right)$ and $\left({T, +, \circ}\right)$ be subrings of $\left({R, +, \circ}\right)$. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\ n}\right]}\right\}$ Then $S T$ is a subring of $\left({R, +, \circ}\right)$.


Smallest Fermat Pseudoprime to Bases 2, 3, 5 and 7 The smallest Fermat pseudoprime to bases $2$, $3$, $5$ and $7$ is $29 \, 341$.


Subset has 2 Conjugates then Normal Subgroup Let $G$ be a group. Let $S$ be a subset of $G$. Let $S$ have exactly two conjugates in $G$. Then $G$ has a proper non-trivial normal subgroup.


Equivalence of Definitions of Complex Inverse Hyperbolic Cotangent {{TFAE|def = Complex Inverse Hyperbolic Cotangent}} Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$


Power Set of Transitive Set is Transitive Let $S$ be a transitive set. Then its power set $\powerset S$ is also a transitive set.


Infimum is not necessarily Smallest Element Let $\struct {S, \preceq}$ be an ordered set. Let $T$ admit a infimum in $S$. Then the infimum of $T$ in $S$ is not necessarily the smallest element of $T$.


Cauchy Sequence is Bounded/Normed Division Ring/Proof 1 Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Every Cauchy sequence in $R$ is bounded.


Bernoulli's Equation/x y^2 y' + y^3 = x cosine x The first order ODE: :$(1): \quad x y^2 y' + y^3 = x \cos x$ has the general solution: :$y^3 = 3 \sin x + \dfrac {9 \cos x} x - \dfrac {18 \sin x} {x^2} - \dfrac {18 \cos x} {x^3} + \dfrac C {x^3}$


Minimum Area of Triangle whose Vertices are Lattice Points Let $T$ be a triangle embedded in a cartesian plane. Let the vertices of $T$ be lattice points which are not all on the same straight line. Then the area of $T$ is such that: :$\map \Area T \ge \dfrac 1 2$


Set of Meet Irreducible Elements Excluded Top is Order Generating Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a continuous complete lattice. Let $X = \mathit{IRR}\left({L}\right) \setminus \left\{ {\top}\right\}$ where $\mathit{IRR}\left({L}\right)$ denotes the set of all meet irreducible element of $S$, :$\top$ denotes the top of $L$. Then $X$ is order generating.


Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable Let $A$ be a class of natural numbers. Let $A$ have no greatest element. Then $A$ is a denumerable class.


Floor defines Equivalence Relation Let $x \in \R$ be a real number. Let $\floor x$ denote the floor function of $x$. Let $\RR$ be the relation defined on $\R$ such that: :$\forall x, y, \in \R: \tuple {x, y} \in \RR \iff \floor x = \floor y$ Then $\RR$ is an equivalence, and $\forall n \in \Z$, the $\RR$-class of $n$ is the half-open interval $\hointr n {n + 1}$.


Sigma-Algebras with Independent Generators are Independent Let $\struct {\Omega, \EE, \Pr}$ be a probability space. Let $\Sigma, \Sigma'$ be sub-$\sigma$-algebras of $\EE$. Suppose that $\GG, \HH$ are $\cap$-stable generators for $\Sigma, \Sigma'$, respectively. Suppose that, for all $G \in \GG, H \in \HH$: :$(1): \quad \map \Pr {G \cap H} = \map \Pr G \map \Pr H$ Then $\Sigma$ and $\Sigma'$ are $\Pr$-independent.


Ordering on Multiindices is Partial Order Let $Z$ be the set of multiindices indexed by a set $J$. The ordering on $Z$ is a partial ordering.


Group Epimorphism Induces Bijection between Subgroups/Corollary Let $H \le G$ denote that $H$ is a subgroup of $G$. Then: :$\forall H \le G, K \subseteq H: \phi \sqbrk H \cong H / K$ where $H / K$ denotes the quotient group of $H$ by $K$.


Power of Two is Even-Times Even Only Let $a > 2$ be a power of $2$. Then $a$ is even-times even only. {{:Euclid:Proposition/IX/32}}


Product of Absolutely Convergent Products is Absolutely Convergent Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely. Let $\displaystyle \prod_{n \mathop = 1}^\infty b_n$ converge absolutely. Then $\displaystyle \prod_{n \mathop = 1}^\infty a_nb_n$ converges absolutely.


Representations for 1 in Golden Mean Number System Then there are infinitely many ways to express the number $1$ in the golden mean number system.


Order 2 Square Matrices with Zero Diagonals do not form Ring Let $S$ be the set of square matrices of order $2$ whose diagonal elements are zero. Then the algebraic structure $\struct {S, +, \times}$ is not a ring. Note that $\times$ denotes conventional matrix multiplication.


Well-Founded Induction Let $\struct {A, \prec}$ be a foundational relation. Let $\prec^{-1} \sqbrk x$ denote the preimage of $x$ for each $x \in A$. Let $B$ be a class such that $B \subseteq A$. Suppose that: :$(1): \quad \forall x \in A: \paren {\prec^{-1} \sqbrk x \subseteq B \implies x \in B}$ Then: :$A = B$ That is, if a property passes from the preimage of $x$ to $x$, then this property is true for all $x \in A$.


Relation Induced by Quotient Set is Equivalence Let $S$ be a set. Let $\mathcal R$ be an equivalence relation on $S$. Let $S / \mathcal R$ be the quotient set of $S$ determined by $\mathcal R$. Let $\mathcal R'$ be the relation induced by $S / \mathcal R$ on $S$. Then $\mathcal R' = \mathcal R$.


Primitive of x squared over Root of a x + b :$\displaystyle \int \frac {x^2 \ \mathrm d x} {\sqrt{a x + b} } = \frac {2 \left({3 a^2 x^2 - 4 a b x + 8 b^2}\right) \sqrt{a x + b} } {15 a^3}$


Linear Second Order ODE/y'' - 2 y' = 12 x - 10 The second order ODE: :$(1): \quad y'' - 2 y' = 12 x - 10$ has the general solution: :$y = C_1 + C_2 e^{2 x} + 2 x - 3 x^2$


2-Digit Permutable Primes The $2$-digit permutable primes are: :$11, 13, 17, 37, 79$ and their anagrams, and no other.


Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers Let $n, k \in \Z_{\ge 0}$ be positive integers. Let $S_k = \displaystyle \sum_{i \mathop = 1}^n i^k$. Then: :$\displaystyle \sum_{i \mathop = 1}^k \binom {k + 1} i S_i = \paren {n + 1}^{k + 1} - \paren {n + 1}$


Construction of Inverse Completion/Cartesian Product with Cancellable Elements Let $\left({S \times C, \oplus}\right)$ be the external direct product of $\left({S, \circ}\right)$ and $\left({C, \circ {\restriction_C}}\right)$, where $\oplus$ is the operation on $S \times C$ induced by $\circ$ on $S$ and $\circ {\restriction_C}$ on $C$. That is: :$\forall \left({x, y}\right), \left({u, v}\right) \in S \times C: \left({x, y}\right) \oplus \left({u, v}\right) = \left({x \circ u, y \mathop{\circ {\restriction_C}} v}\right)$ Then $\left({S \times C, \oplus}\right)$ is a commutative semigroup.


Simple Variable End Point Problem/Endpoints on Curves Let $y$, $F$, $\phi$ and $\psi$ be smooth real functions. Let $J = J \sqbrk y$ be a functional of the form: :$\ds J \sqbrk y = \int_{x_0}^{x_1} \map F {x, y, y'} \rd x$ Let $P_0$, $P_1$ be the endpoints of the curve $y$. Suppose $P_0$, $P_1$ lie on curves $y = \map {\phi} x$, $y = \map {\psi} x$. Then the extremum of $J \sqbrk y$ is a curve which satisfies the following system of Euler and transversality equations: {{begin-eqn}} {{eqn | l = F_y - \dfrac {\d} {\d x} F_{y'}       | r = 0 }} {{eqn | l = \bigvalueat {\paren {F + \paren {\phi' - y'} F_{y'} } } {x \mathop = x_0}       | r = 0 }} {{eqn | l = \bigvalueat {\paren {F + \paren {\psi' - y'} F_{y'} } } {x \mathop = x_1} = 0       | r = 0 }} {{end-eqn}} {{explain|meaning of square brackets}}


Sample Matrix Independence Test Let $V$ be a vector space of real or complex-valued functions on a set $J$. Let $f_1, \ldots, f_n$ be functions in $V$. Let '''samples''' $x_1, \ldots, x_n$ from $J$ be given. Define '''sample matrix'''  :$\displaystyle S = \begin{bmatrix} f_1(x_1)   &  \cdots      & f_n(x_1) \\ \vdots     & \ddots       & \vdots \\ f_1(x_n)   & \cdots       & f_n(x_n) \\ \end{bmatrix}$ Let $S$ be invertible. Then $f_1, \ldots, f_n$ are linearly independent in $V$.


Difference of Images under Mapping not necessarily equal to Image of Difference Let $f: S \to T$ be a mapping. The image of the set difference of two subsets of $S$ is not necessarily equal to the set difference of the images. That is: Let $S_1$ and $S_2$ be subsets of $S$. Then it is not always the case that: :$f \sqbrk {S_1} \setminus f \sqbrk {S_2} = f \sqbrk {S_1 \setminus S_2}$ where $\setminus$ denotes set difference.


Equation of Cissoid of Diocles/Cartesian Form The cissoid of Diocles can be defined by the Cartesian equation: :$x \paren {x^2 + y^2} = 2 a y^2$


Ordinal Exponentiation via Cantor Normal Form/Corollary Let $x$ and $y$ be ordinals. Let $x$ be a limit ordinal and let $y > 0$. Let $\langle a_i \rangle$ be a sequence of ordinals that is strictly decreasing on $1 \le i \le n$. Let $\langle b_i \rangle$ be a sequence of natural numbers. Then: :$\displaystyle \left({ \sum_{i \mathop = 1}^n x^{a_i} \times b_i }\right)^{x^y} = x^{a_1 \mathop \times x^y}$ {{explain|what "Cantor normal form" is in the exposition of the above.}}


Arens-Fort Space is Non-Meager Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is a non-meager space.


Chu-Vandermonde Identity for Gaussian Binomial Coefficients {{begin-eqn}} {{eqn | l = \binom {r + s} n_q       | r = \sum_k \binom r k_q \binom s {n - k}_q q^{\left({r - k}\right) \left({n - k}\right)}       | c =  }} {{eqn | r = \sum_k \binom r k_q \binom s {n - k}_q q^{\left({s - n + k}\right) k}       | c =  }} {{end-eqn}}


Conditions for Homogeneity/Straight Line The line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ is homogeneous {{iff}} $\beta = 0$.


Sum of Two Rational 4th Powers but not Two Integer 4th Powers $5906$ is the smallest integer which can be expressed as the sum of two rational $4$th powers, but not two integer $4$th powers.


Uncountable Finite Complement Topology is not Perfectly T4 Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$. Then $T$ is not a perfectly $T_4$ space.


Fibonacci Numbers which equal their Index The only Fibonacci numbers which equal their index are: {{begin-eqn}} {{eqn | l = F_0       | r = 0 }} {{eqn | l = F_1       | r = 1 }} {{eqn | l = F_5       | r = 5 }} {{end-eqn}}


Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples Let $n$ be a two-digit positive integer with the following property: :Let $m$ be a $3$-digit multiple of $n$. :Then any cyclic permutation of the digits of $m$ is also a multiple of $n$. Then $n$ is either $27$ or $37$.


Existence of Homomorphism between Localizations of Ring at Elements Let $A$ be a commutative ring with unity. Let $f, g \in A$. {{TFAE}} :$(1): \quad$ There exists an $A$-algebra homomorphism $h : A_f \to A_g$ between localizations, the '''induced homomorphism'''. :$(2): \quad f$ divides some power of $g$. :$(3): \quad$ There is an inclusion of vanishing sets: $\map V f \subseteq \map V g$. That is, every prime ideal containing $f$ also contains $g$. :$(4): \quad$ There is an inclusion of principal open subsets: $\map D f \supseteq \map D g$


Arcsecant Logarithmic Formulation Let $x$ be a real number.  Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$. Then:  :$\displaystyle \arcsec x = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}$ where: :$\arcsec$ is the arcsecant function :$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$.


P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic Numbers as the Quotient of Cauchy Sequences. Then the sequence of partial sums of the series: :$\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ represents an element of $\struct {\Q_p,\norm {\,\cdot\,}_p}$.


Position of Cart attached to Wall by Spring under Damping/Overdamped/x = x0 at t = 0 Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = \dfrac {x_0} {m_1 - m_2} \left({m_1 e^{m_2 t} - m_2 e^{m_1 t} }\right)$


Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex Let $f$ be a real function which is strictly convex on the open interval $I$. Let $J = f \sqbrk I$. If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly convex on $J$.


Measure Space from Outer Measure Suppose $\mu^*$ is an outer measure on a set $X$. Let $\mathfrak M(\mu^*)$ be the collection of $\mu^*$-measurable sets. Let $\mu$ be the restriction of $\mu^*$ to $\mathfrak M(\mu^*)$. Then $(X, \mathfrak M(\mu^*), \mu)$ is a measure space.


Summation over k of Ceiling of mk+x over n Let $m, n \in \Z$ such that $n > 0$. Let $x \in \R$. Then: :$\displaystyle \sum_{0 \mathop \le k \mathop < n} \left \lceil{\dfrac {m k + x} n}\right \rceil = \dfrac {\left({m + 1}\right) \left({n - 1}\right)} 2 - \dfrac {d - 1} 2 + d \left \lceil{\dfrac x d}\right \rceil$ where: :$\left \lceil{x}\right \rceil$ denotes the ceiling of $x$ :$d$ is the greatest common divisor of $m$ and $n$.


Pointwise Minimum of Measurable Functions is Measurable Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions. Then the pointwise minimum $\min \left({f, g}\right): X \to \overline{\R}$ is also $\Sigma$-measurable.


Numbers between which exists one Mean Proportional are Similar Plane Let $a, b \in \Z$ such that the geometric mean is an integer. Then $a$ and $b$ are similar plane numbers. {{:Euclid:Proposition/VIII/20}}


Closure in Infinite Particular Point Space is not Compact Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Let $A \in \tau_p$ be open in $T$. Let $A^-$ be the closure of $A$. Then $A^-$ is not compact.


Reciprocal of 8 The reciprocal of $8$ can be expressed as the summation of the powers of $2$ multiplied by the reciprocal  powers of $10$: :$\dfrac 1 8 = \displaystyle \sum_{k \mathop \ge 0} \dfrac {2^k} {10^{k + 1} }$ <pre> .1   2    4     8     16      32       64       128        256         512         1024          .... ------------- .12499999744 ------------- </pre>


Singleton is Independent implies Rank is One/Corollary :$\set x$ is an independent subset {{iff}} $\map \rho {\set x} = 1$


Derivative of Hyperbolic Secant Function :$\map {\dfrac \d {\d x} } {\sech u} = -\sech u \tanh u \dfrac {\d u} {\d x}$


Min Semigroup on Toset is Semilattice Let $\struct {S, \preceq}$ be a totally ordered set.  Then the min semigroup $\struct {S, \min}$ is a semilattice.


Sum over k to n of Stirling Number of the Second Kind of k with m by m+1^n-k Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_{k \mathop \le n} {k \brace m} \paren {m + 1}^{n - k} = {{n + 1} \brace {m + 1}}$ where $\displaystyle {k \brace m}$ etc. denotes a Stirling number of the second kind.


Congruence Relation on Ring induces Ring Let $\struct {R, +, \circ}$ be a ring. Let $\EE$ be a congruence relation on $R$ for both $+$ and $\circ$. Let $R / \EE$ be the quotient set of $R$ by $\EE$. Let $+_\EE$ and $\circ_\EE$ be the operations induced on $R / \EE$ by $+$ and $\circ$ respectively. Then $\struct {R / \EE, +_\EE, \circ_\EE}$ is a ring.


Power Series Expansion for Exponential of Sine of x :$e^{\sin x} = 1 + x + \dfrac {x^2} 2 - \dfrac {x^4} 8 - \dfrac {x^5} {15} + \cdots$ for all $x \in \R$.


Proper Filter is Included in Ultrafilter in Boolean Lattice Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a Boolean lattice. Let $F$ be a proper filter on $L$. Then there exists ultrafilter $G$ on $L$: $F \subseteq G$


Integer as Sum of 5 Non-Zero Squares Let $n \in \Z$ be an integer such that $n > 33$. Then $n$ can be expressed as the sum of $5$ non-zero squares.


Equidistance is Independent of Betweenness Let $\mathcal G$ be a formal systematic treatment of geometry containing only: : The language and axioms of first-order logic, and the disciplines preceding it : The undefined terms of Tarski's Geometry (excluding equidistance)  : Some or all of Tarski's Axioms of Geometry. In $\mathcal G$, equidistance $\equiv$ is necessarily an undefined term with respect to betweenness $\mathsf B$.


Existence of Interval of Convergence of Power Series/Corollary 2 Let $\displaystyle S \left({x}\right) = \sum_{n \mathop = 0}^\infty a_n x^n$ be a power series. Let $S \left({x}\right)$ be convergent at $x = x_0$. Then $S \left({x}\right)$ is convergent for all $x$ such that $\left\lvert{x}\right\rvert < \left\lvert{x_0}\right\rvert$.


Equation of Line in Complex Plane/Formulation 2 Let $\C$ be the complex plane. Let $L$ be the infinite straight line in $\C$ which is the locus of the equation: :$l x + m y = 1$ Then $L$ may be written as: :$\map \Re {a z} = 1$ where $a$ is the point in $\C$ defined as: :$a = l - i m$


Epimorphism into Projective Object Splits Let $\mathbf C$ be a metacategory. Let $P \in \mathbf C_0$ be a projective object of $\mathbf C$. Let $e: E \twoheadrightarrow P$ be an epimorphism. Then $e$ is a split epimorphism, i.e. it admits a retraction $f: P \to E$.


Derivative of Exponential of Function :$\map {\dfrac \d {\d x} } {e^u} = e^u \dfrac {\d u} {\d x}$


Taxicab Metric on Real Vector Space is Metric The taxicab metric on the real vector space $\R^n$ is a metric.


Commutative B-Algebra is Entropic Structure Let $\left({G, *}\right)$ be a commutative $B$-algebra. Then $\left({G, *}\right)$ is an entropic structure.


Substitution in Big-O Estimate/General Result Let $X$ and $Y$ be topological spaces. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\,\cdot\,}$. Let $x_0 \in X$ and $y_0 \in Y$. Let $f: X \to Y$ be a function with $\map f {x_0} = y_0$ that is continuous at $x_0$. Let $g, h: Y \to V$ be functions. Suppose $\map g y = \map O {\map h y}$ as $y \to y_0$, where $O$ denotes big-O notation. Then $\map {\paren {g \circ f} } x = \map O {\map {\paren {h \circ f} } x}$ as $x \to x_0$.


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


First Order ODE/(x^2 - 2 y^2) dx + x y dy = 0 is a homogeneous differential equation with solution: :$y^2 = x^2 + C x^4$


Relative Prime Modulo Tensor is Zero Let $p \in \Z_{>0}$ and $q \in \Z_{>0}$ be positive coprime integers. Let $\Z / p \Z$ and $\Z / q \Z$ be $\Z$-modules. {{explain|It is not a good idea to use the same notation for both a ring and a module. Either $\Z / p \Z$ is a ring or it is a module. Please consider taking the advice in the explain template at the bottom of this page.}} Then: :$\Z / p \Z \otimes_\Z \Z / q\Z = 0$ where $\otimes_\Z$ denotes tensor product over integers.


Congruence of Sum with Constant Let $a, b, z \in \R$. Let $a$ be congruent to $b$ modulo $z$: : $a \equiv b \pmod z$ Then: :$\forall c \in \R: a + c \equiv b + c \pmod z$


Infinite Ramsey's Theorem implies Finite Ramsey's Theorem :$\forall l, n, r \in \N: \exists m \in \N: m \to \left({l}\right)_r^n$ where $\alpha \to \left({\beta}\right)^n_r$ means that: :for any assignment of $r$-colors to the $n$-subsets of $\alpha$ ::there is a particular color $\gamma$ and a subset $X$ of $\alpha$ of size $\beta$ such that all $n$-subsets of $X$ are $\gamma$.


Equivalent Subobjects have Isomorphic Domains Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Let $m, m'$ be equivalent subobjects of $C$. Then there exists an isomorphism $f: m \to m'$.


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


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


Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x :$\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }$


Primitive of Cube of Cosecant of a x :$\ds \int \csc^3 a x \rd x = \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C$


Complement of Complete Graph is Edgeless Graph Let $K_n$ denote the complete graph of order $n$. Then the complement of $K_n$ is the $n$-edgeless graph $N_n$.


Anning's Theorem In any base greater than $1$, the fraction: :$\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$ has the property that if the two $1$'s in the center of the numerator and the denominator are replaced by the same odd number of $1$'s, the value of the fraction remains the same. For example: :$\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011} = \dfrac {1 \, 010 \, 111 \, 110 \, 101} {1 \, 100 \, 111 \, 110 \, 011} = \dfrac {9091} {9901}$ (in base $10$).


Summation of Zero/Set Let $S$ be a set. Let $0: S \to \mathbb A$ be the zero mapping. Then the summation with finite support of $0$ over $S$ equals zero: :$\displaystyle \sum_{s \mathop \in S} \map 0 s = 0$


Subset of Indiscrete Space is Everywhere Dense Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$ such that $H \ne \O$. Then $H$ is everywhere dense.


Inverse Cotangent of Imaginary Number :$\cot^{-1} \paren {i x} = - i \coth^{-1} x$


Primitive of Power of Root of a x + b :$\displaystyle \int \left({\sqrt{a x + b} }\right)^m \ \mathrm d x = \frac {2 \left({\sqrt{a x + b} }\right)^{m + 2} } {a \left({m + 2}\right)} + C$


Transpose of Linear Transformation is a Linear Transformation Let $R$ be a commutative ring. Let $G$ and $H$ be $R$-modules. Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively. Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Let $u \in \map {\LL_R} {G, H}$. Let $u^t: H^* \to G^*$ be the transpose of $u$. Then $u^t: H^* \to G^*$ is itself a linear transformation.


Convergence of P-Series/Divergence if p between 0 and 1 Let $0 < \map \Re p \le 1$. Then the $p$-series: :$\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$ diverges.


Distance between Two Parallel Straight Lines is Everywhere the Same Let $AB$ and $CD$ be parallel straight lines. Let perpendiculars $EF$ and $GH$ be drawn from $AB$ to $CD$, where $E, G$ are on $AB$ and $F, H$ are on $CD$. Then $EF = GH$. That is, the distance between  $AB$ and $CD$ is the same everywhere along their length.


Union Distributes over Intersection/Family of Sets/Corollary Let $I$ and $J$ be indexing sets. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$. Then: :$\displaystyle \bigcap_{\tuple{\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}$ where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Order Topology on Natural Numbers is Discrete Topology Let $\le$ be the standard ordering on the natural numbers $\N$. Then the order topology $\tau$ on $\N$ is the discrete topology.


Set with Two Parallel Elements is Dependent Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x, y \in S$. Let $x, y$ be parallel elements. If $x, y \in A$ then $A$ is dependent.


Klein Four-Group as Order 2 Matrices Let $G$ be the set of order $2$ square matrices: :$G = \set {I, A, B, C}$ where: :$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ Then the algebraic structure $\struct {G, \times}$, where $\times$ denotes (conventional) matrix multiplication, forms the Klein four-group.


Conjunction Equivalent to Negation of Implication of Negative/Formulation 1/Proof :$p \land q \dashv \vdash \neg \left({p \implies \neg q}\right)$


Leibniz's Formula for Pi :$\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$


Indiscrete Space is Hereditarily Compact Let $\struct {S, \tau}$ be an indiscrete topological space. Then $\struct {S, \tau}$ is hereditarily compact.


Primitive of Power of x by Sine of a x :$\displaystyle \int x^m \sin a x \rd x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x$


Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element Every bounded subset of $N$ has a greatest element.


Pentagonal and Hexagonal Numbers The sequence of positive integers which are simultaneously pentagonal and hexagonal begins: :$1, 40 \, 755, 1 \, 533 \, 776 \, 805, 57 \, 722 \, 156 \, 241 \, 751, \ldots$ {{OEIS|A046180}}


Principle of Counting Let $T$ be a set such that $T \sim \N_n$. Then: : $\forall m \in \N: n \ne m \implies T \nsim \N_m$


Hilbert's Basis Theorem/Corollary Let $A$ be a Noetherian ring. Let $n \ge 1$ be an integer. Let $A \sqbrk {x_1, \ldots, x_n}$ be the ring of polynomial forms over $A$ in the indeterminates $x_1, \ldots, x_n$. Then $A \sqbrk {x_1, \ldots, x_n}$ is also a Noetherian ring.


Existence and Uniqueness of Direct Limit of Sequence of Groups Let $\sequence {G_n}_{n \mathop \in \N}$ be a sequence of groups. Let $\sequence {g_n}_{n \mathop \in \N}: g_n: G_n \to G_{n + 1}$ be a sequence of group homomorphisms. Then their direct limit $G_\infty$ exists and is unique up to unique isomorphism.


Integral to Infinity of Dirac Delta Function Let $\map \delta x$ denote the Dirac delta function. Then: :$\displaystyle \int_0^{+ \infty} \map \delta x \rd x = 1$


Condition for Continuity on Interval Let $f$ be a real function defined on an interval $\mathbb I$. Then $f$ is continuous on $\mathbb I$ {{iff}}: :$\forall x \in \mathbb I: \forall \epsilon > 0: \exists \delta > 0: y \in \mathbb I \land \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon$


Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact Let $T = \left({S, \tau}\right)$ be a weakly locally compact topological space. Let $M = \left({A, d}\right)$ be a metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n : X\to M$. Let $f_n$ converge compactly to $f:X\to M$. Then $f_n$ converges locally uniformly to $f$.


Gamma Function of Minus 5 over 2 :$\map \Gamma {-\dfrac 5 2} = -\dfrac {8 \sqrt \pi} {15}$


Rational Power is of Exponential Order Epsilon Let $r = \dfrac p q$ be a rational number, with $p, q \in \Z: q \ne 0, r > 0$. Then:  :$t \mapsto t^r$ is of exponential order $\epsilon$ for any  $\epsilon > 0$ arbitrarily small in magnitude.


Countable Complement Space is Connected Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is a connected space.


Power of 2 is Almost Perfect Let $n \in \Z_{>0}$ be a power of $2$: :$n = 2^k$ for some $k \in \Z_{>0}$. Then $n$ is almost perfect.


Cardinality of Finite Set is Well-Defined Let $S$ be a finite set. Then there is a unique natural number $n$ such that $S \sim \N_n$, where: :$\sim$ represents set equivalence and: :$\N_n = \set {0, 1, \dotsc, n - 1}$ is the initial segment of $\N$ determined by $n$.


Temperature of Body under Newton's Law of Cooling Let $B$ be a body in an environment whose ambient temperature is $H_a$. Let $H$ be the temperature of $B$ at time $t$. Let $H_0$ be the temperature of $B$ at time $t = 0$. Then: :$H = H_a - \paren {H_0 - H_a} e^{-k t}$ where $k$ is some positive constant.


Floor Function/Examples/Floor of Root 2 :$\floor {\sqrt 2} = 1$


Group Action on Subgroup by Left Regular Representation Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $*: H \times G \to G$ be the operation defined as: :$\forall \left({h, g}\right) \in H \times G: h * g = \lambda_h \left({g}\right)$ where $\lambda_h \left({g}\right)$ is the left regular representation of $g$ by $h$. Then $*$ is a group action.


Composition of Affine Transformations is Affine Transformation Let $\EE$, $\FF$ and $\GG$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively. Let $\LL: \EE \to \FF$ and $\MM: \FF \to \GG$ be affine transformations. Let $L$ and $M$ be the tangent maps of $\LL$ and $\MM$ respectively. Then the composition $\MM \circ \LL: \EE \to \FF$ is an affine transformation with tangent map $M \circ L$.


Derivative of Periodic Function Let $f: \R \to \R$ be a real function. Let $f$ be differentiable on all of $\R$. Then if $f$ is periodic with period $L$, then its derivative is also periodic with period $L$.


Integral to Infinity of Dirac Delta Function by Continuous Function Let $\map \delta x$ denote the Dirac delta function. Let $g$ be a continuous real function. Then: :$\displaystyle \int_0^{+ \infty} \map \delta x \, \map g x \rd x = \map g 0$


Diagonal Relation is Many-to-One The diagonal relation is many-to-one. That is: :$\forall x \in \Dom {\Delta_S}: \tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S \implies y_1 = y_2$ where $\Delta_S$ is the diagonal relation on a set $S$.


Zero of Power Set with Union Let $S$ be a set and let $\powerset S$ be its power set. Consider the algebraic structure $\struct {\powerset S, \cup}$, where $\cup$ denotes set union. Then $S$ serves as the zero element for $\struct {\powerset S, \cup}$.


Characterization of Derivative by Local Basis Let $T = \struct {S, \tau}$ be a topological space. Let $A$ be a subset of $S$. Let $x$ be a point of $T$. Let $\BB \subseteq \tau$ be a local basis at $x$. Then :$x \in A'$ {{iff}}: :for every $U \in \BB$, there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$ where: :$A'$ denotes the derivative of $A$.


Triangular Matrices forms Subring of Square Matrices Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map {\mathcal M_R} n$ be the order $n$ square matrix space over a ring $R$. Let $\struct {\map {\mathcal M_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Let $\map {U_R} n$ be the set of upper triangular matrices of order $n$ over $R$. Then $\map {U_R} n$ forms a subring of $\struct {\map {\mathcal M_R} n, +, \times}$. Similarly, let $\map {L_R} n$ be the set of lower triangular matrices of order $n$ over $R$. Then $\map {L_R} n$ forms a subring of $\struct {\map {\mathcal M_R} n, +, \times}$.


Laplace Transform of Identity Mapping Let $\laptrans f$ denote the Laplace transform of a function $f$. Let $\map {I_\R} t$ denote the identity mapping on $\R$ for $t > 0$. Then: :$\laptrans {\map {I_\R} t} = \dfrac 1 {s^2}$ for $\map \Re s > 0$.


Linear Combination of Mellin Transforms Let $\mathcal M$ be the Mellin transform. Let $f \left({t}\right)$, $g \left({t}\right)$ be functions such that $\mathcal M \left\{ {f \left({t}\right)}\right\} \left({s}\right)$ and $\mathcal M \left\{ {f \left({t}\right)}\right\} \left({s}\right)$ exist. Let $\lambda \in \C$ be a constant. Then: :$\mathcal M \left\{ {\lambda f \left({t}\right) + g \left({t}\right)}\right\} \left({s}\right) = \lambda \mathcal M \left\{ {f \left({t}\right)}\right\}\left({s}\right) + \mathcal M \left\{ {g \left({t}\right)}\right\} \left({s}\right)$ everywhere all the above expressions are defined.


Ordering on Mappings Implies Galois Connection Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $g: S \to T$ and $d: T \to S$ be mappings such that: :$g$ and $d$ are increasing mappings  and :$d \circ g \preceq I_S$ and $I_T \precsim g \circ d$ Then :$\left({g, d}\right)$ is Galois connection. where :$\preceq, \precsim$ denote the orderings on mappings :$I_S$ denotes the identity mapping of $S$ :$\circ$ denotes the composition of mappings.


Product of Divisors is Divisor of Product Let $a, b, c, d \in \Z$ be integers such that $a, c \ne 0$. Let $a \divides b$ and $c \divides d$, where $\divides$ denotes divisibility. Then: :$a c \divides b d$


Metric Space Completeness is not Preserved by Homeomorphism Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be a homeomorphism. If $M_1$ is complete then it is not necessarily the case that so is $M_2$.


Effect of Elementary Column Operations on Determinant Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Take the elementary column operations: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda$, multiply column $i$ by $\lambda$         | m = \kappa_i \to \lambda \kappa_i }} {{axiom | n = \text {ECO} 2         | t = For some $\lambda$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{axiom | n = \text {ECO} 3         | t = Exchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}} Applying $\text {ECO} 1$ has the effect of multiplying $\map \det {\mathbf A}$ by $\lambda$. Applying $\text {ECO} 2$ has no effect on $\map \det {\mathbf A}$. Applying $\text {ECO} 3$ has the effect of multiplying $\map \det {\mathbf A}$ by $-1$.


Biconditional as Disjunction of Conjunctions/Formulation 2/Proof 2 : $\vdash \left({p \iff q}\right) \iff \left({\left({p \land q}\right) \lor \left({\neg p \land \neg q}\right)}\right)$


Join of Sets of Integer Multiples is Set of Integer Multiples of GCD Let $m, n \in \Z$. Let $m \Z$ denote the set of integer multiples of $m$ Let $r \in \Z$ such that: :$m \Z \subseteq r \Z$ and: :$n \Z \subseteq r \Z$ Then: :$\gcd \set {m, n} \Z \subseteq r \Z$ where $\gcd$ denotes greatest common divisor.


Euler's Equation/Integrated wrt Length Element Let $y$ be a real mapping belonging to $C^2$ differentiability class. Assume that: :$\ds J \sqbrk y = \int_a^b \map f {x, y, y'} \rd s$ where :$\rd s = \sqrt {1 + y'^2} \rd x$ Then Euler's Equation can be reduced to: :$f_y - f_x y' - f_{y'} y' y'' - f \dfrac {y''} {\paren {1 + y'^2}^{\frac 3 2} } = 0$


Well-Ordered Induction Let $\left({A,\prec}\right)$ be a strict well-ordering. For all $x \in A$, let the $\prec$-initial segment of $x$ be a small class. Let $B$ be a class such that $B \subseteq A$. Let: :$(1): \quad \forall x \in A: \left({ \left({ A \cap \prec^{-1} \left({ x }\right) }\right) \subseteq B \implies x \in B }\right)$ Then: :$A = B$ That is, if a property passes from the initial segment of $x$ to $x$, then this property is true for all $x \in A$.


Regiomontanus' Angle Maximization Problem Let $AB$ be a line segment. Let $AB$ be produced to $P$. Let $PQ$ be constructed perpendicular to $AB$. Then the angle $AQB$ is greatest when $PQ$ is tangent to a circle passing through $A$, $B$ and $Q$.


Zero Vector is Unique Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms. Then the zero vector in $\mathbf V$ is unique: :$\exists! \mathbf 0 \in \mathbf V: \forall \mathbf x \in \mathbf V: \mathbf x + \mathbf 0 = \mathbf x$


Either-Or Topology is not Separable Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a separable space.


Range of Characters Let $G$ be a finite abelian group of order $m$. Let $\chi: G \to \C^\times$ be a character on $G$. Then for any $g \in G$, $\map \chi g$ is an $m$th root of unity. If $e$ is the identity of $G$ then $\map \chi g = 1$.


Lexicographic Order Initial Segments Let $\operatorname{Le}$ denote the lexicographic order for the set $\left({\operatorname{On} \times \operatorname{On} }\right)$. Let the ordinal number $1$ denote the successor of $\varnothing$. Then the initial segment of $ \left({1, \varnothing}\right)$ with respect to the lexicographic order $\operatorname{Le}$ is a proper class. This initial segment shall be denoted $\left({\operatorname{On} \times \operatorname{On} }\right)_{\left({1, \varnothing}\right)}$. {{explain|The connection between these two statements is not clear.}}


Limit Points of Sequence in Indiscrete Space on Uncountable Set Let $S$ be an uncountable set. Let $T = \struct {S, \set {\O, S} }$ be the indiscrete topological space on $S$. Let $\sequence {s_n}$ be a sequence in $T$. Then every sequence in $T$ has an uncountable number of limit points.


Equivalence of Definitions of Symmetric Difference Let $S$ and $T$ be sets. {{TFAE|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}


Open Real Interval is Subset of Closed Real Interval Let $a, b \in \R$ be real numbers. Then: :$\openint a b \subseteq \closedint a b$ where: :$\openint a b$ is the open interval between $a$ and $b$ :$\closedint a b$ is the closed interval between $a$ and $b$.


Relation Isomorphism Preserves Equivalence Relations Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be relational structures. Let $\struct {S, \RR_1}$ and $\struct {T, \RR_2}$ be (relationally) isomorphic. Then $\RR_1$ is an equivalence relation {{iff}} $\RR_2$ is also an equivalence relation.


Sum over k of -2 Choose k :$\displaystyle \sum_{k \mathop = 0}^n \binom {-2} k = \left({-1}\right)^n \left\lceil {\dfrac {n + 1} 2}\right\rceil$ where: :$\dbinom {-2} k$ is a binomial coefficient :$\left\lceil {x}\right\rceil$ denotes the ceiling of $x$.


Composite of Epimorphisms is Epimorphism Let: :$\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$ :$\struct {S_2, *_1, *_2, \ldots, *_n}$ :$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be algebraic structures. Let: :$\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$ :$\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ be epimorphisms. Then the composite of $\phi$ and $\psi$ is also an epimorphism.


Axiom of Choice Implies Zorn's Lemma Acceptance of the Axiom of Choice implies the truth of Zorn's Lemma.


Norm of Eisenstein Integer Let $\alpha$ be an Eisenstein integer. That is, $\alpha = a + b \omega$ for some $a, b \in \Z$, where $\omega = e^{2\pi i /3}$. Then: :$\cmod \alpha^2 = a^2 - a b + b^2$ where $\cmod {\, \cdot \,}$ denotes the modulus of a complex number.


Conditions on Rational Solution to Polynomial Equation Let $P$ be the polynomial equation: :$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$ where $a_0, \ldots, a_n$ are integers. Let $\dfrac p q$ be a root of $P$ expressed in canonical form. Then $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$.


Negated Upper Index of Binomial Coefficient/Complex Numbers For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers: :$\dbinom z w = \dfrac {\sin \left({\pi \left({w - z - 1}\right)}\right)} {\sin \left({\pi z}\right)} \dbinom {w - z - 1} w$ where $\dbinom z w$ is a binomial coefficient.


Hamiltonian of Standard Lagrangian is Total Energy Let $P$ be a physical system of classical particles. Let $L$ be a standard Lagrangian associated with $P$. Then the Hamiltonian of $P$ is the total energy of $P$.


Normal to Cycloid passes through Bottom of Generating Circle Let $C$ be a cycloid generated by the equations: : $x = a \left({\theta - \sin \theta}\right)$ : $y = a \left({1 - \cos \theta}\right)$ Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$.


Argument of x to the n Equals n Times The Argument Let $z$ be a complex number. Then: :$\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$


Permutation is Product of Transpositions Let $S_n$ denote the symmetric group on $n$ letters. Every element of $S_n$ can be expressed as a product of transpositions.


Idempotent Elements for Integer Multiplication There are exactly two integers which are idempotent with respect to multiplication: :$0 \times 0 = 0$ :$1 \times 1 = 1$


Zero Vector is Linearly Dependent Let $G$ be a group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module. Then the singleton set $\set e$ consisting of the zero vector is linearly dependent.


Power Set with Union and Intersection forms Boolean Algebra Let $S$ be a set, and let $\powerset S$ be its power set. Denote with $\cup$, $\cap$ and $\complement$ the operations of union, intersection and complement on $\powerset S$, respectively. Then $\struct {\powerset S, \cup, \cap, \complement}$ is a Boolean algebra.


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


Diagonal Relation is Universally Congruent The diagonal relation $\Delta_S$ on a set $S$ is universally congruent with every operation on $S$.


Discrete Space is Totally Disconnected Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$. Then $T$ is totally disconnected.


Sine of Integer Multiple of Argument/Formulation 2 {{begin-eqn}} {{eqn | l = \sin n \theta       | r = \cos^n \theta \paren {\paren {\tan \theta} - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}       | c =  }} {{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k + 1} \paren {\tan^{2 k + 1} \theta}       | c =  }} {{end-eqn}}


Taylor Series of Logarithm of Gamma Function Let $\gamma$ denote the Euler-Mascheroni constant. Let $\map \zeta s$ denote the Riemann zeta function. Let $\map \Gamma z$ denote the gamma function. Let $\Log$ denote the natural logarithm. Then $\map \Log {\map \Gamma z}$ has the power series expansion: {{begin-eqn}} {{eqn | l = \map \Log {\map \Gamma z}       | r = -\map \gamma {z - 1} + \sum_{k \mathop = 2}^\infty \frac {\paren {-1}^k \map \zeta k} k \paren {z - 1}^k  }} {{end-eqn}} which is valid for all $z \in \C$ such that $\cmod {z - 1} < 1$.


Space is First-Countable iff Character not greater than Aleph 0 Let $T$ be a topological space. $T$ is first-countable {{iff}}: :$\chi \left({T}\right) \leq \aleph_0$ where $\chi \left({T}\right)$ denotes the character of $T$.


Primitive of Reciprocal of x squared by x fourth plus a fourth :$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^4 + a^4}\right)} = \frac {-1} {a^4 x} - \frac {-1} {4 a^5 \sqrt 2} \ln \left({\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} }\right) + \frac 1 {2 a^5 \sqrt 2} \left({\arctan \left({1 - \frac {x \sqrt 2} a}\right) - \arctan \left({1 + \frac {x \sqrt 2} a}\right)}\right)$


Condition for Independence of Discrete Random Variables Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Then $X$ and $Y$ are independent {{iff}} there exist functions $f, g: \R \to \R$ such that the joint mass function of $X$ and $Y$ satisfies: :$\forall x, y \in \R: \map {p_{X, Y} } {x, y} = \map f x \map g y$


Integers under Multiplication do not form Group The set of integers under multiplication $\struct {\Z, \times}$ does not form a group.


Finite Union of Bounded Subsets Let $M = \left({A, d}\right)$ be a metric space. Then the union of any finite number of bounded subsets of $M$ is itself bounded.


First Order ODE/x^2 y' - 3 x y - 2 y^2 = 0 is a homogeneous differential equation with solution: :$y = C x^2 \paren {x + y}$


Intersection of Exteriors contains Exterior of Union Let $T$ be a topological space. Let $\mathbb H$ be a set of subsets of $T$. That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then: :$\displaystyle \paren {\bigcup_{H \mathop \in \mathbb H} H}^e \subseteq \bigcap_{H \mathop \in \mathbb H} H^e $ where $H^e$ denotes the exterior of $H$.


Cycloid has Tautochrone Property Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line. Let a bead $B$ be released from some point on the wire. The time taken for $B$ to reach the lowest point of $C$ is: :$T = \pi \sqrt {\dfrac a g}$ independently of the point at which $B$ is released from. That is, a cycloid is a tautochrone.


Ratio of 2016 to Aliquot Sum $2016$ has the property that its ratio to its aliquot sum is $4 : 9$.


Natural Number is Union of its Successor Let $n \in \N$ be a natural number as defined by the von Neumann construction. Then: :$\map \bigcup {n^+} = n$


Smallest Positive Integer Combination is Greatest Common Divisor Let $a, b \in \Z_{>0}$ be (strictly) positive integers. Let $d \in \Z_{>0}$ be the smallest positive integer such that: : $d = a s + b t$ where $s, t \in \Z$. Then: :$(1): \quad d \divides a \land d \divides b$ :$(2): \quad c \divides a \land c \divides b \implies c \divides d$ where $\divides$ denotes divisibility. That is, $d$ is the greatest common divisor of $a$ and $b$.


Infinite Number of Even Fermat Pseudoprimes Despite their relative rarity, there exist an infinite number of even Fermat pseudoprimes.


Fibonacci Number of Odd Index by Golden Mean Modulo 1 Let $n \in \Z$ be an integer. Then: :$F_{2 n + 1} \phi \bmod 1 = \phi^{-2 n - 1}$ where: :$F_n$ denotes the $n$th Fibonacci number :$\phi$ is the golden mean: $\phi = \dfrac {1 + \sqrt 5} 2$


Integral of Positive Simple Function Well-Defined Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \R, f \in \mathcal{E}^+$ be a positive simple function. Then the $\mu$-integral of $f$, $I_\mu \left({f}\right)$, is well-defined. That is, for any two standard representations for $f$, say: :$\displaystyle f = \sum_{i \mathop = 0}^n a_i \chi_{E_i} = \sum_{j \mathop = 0}^m b_j \chi_{F_j}$ it holds that: :$\displaystyle \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right) = \sum_{j \mathop = 0}^m b_j \mu \left({F_j}\right)$


Subrings of Integers are Sets of Integer Multiples/Examples/Even Integers Let $2 \Z$ be the set of even integers. Then $\struct {2 \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$.


Linear Second Order ODE/y'' + 8 y = 0 The second order ODE: :$(1): \quad y'' + 8 y = 0$ has the general solution: :$y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$


Primitive of Reciprocal of x by Half Integer Power of a x squared plus b x plus c Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }$


Three Times Sum of Cubes of Three Indeterminates Plus 6 Times their Product :$3 \paren {a^3 + b^3 + c^3 + 6 a b c} = \paren {a + b + c}^3 + \paren {a + b \omega + c \omega^2}^3 + \paren {a + b \omega^2 + c \omega}^3$ where: : $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$


Morphism in Preorder Category is Epic Let $\mathbf P$ be a preorder category. Let $f \in \mathbf P_1$ be a morphism. Then $f$ is epic.


Topologies on Set with More than One Element may not be Homeomorphic Let $S$ be a set which contains at least $2$ elements. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Then it is not necessarily the case that $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic.


Either-Or Topology is Scattered Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a scattered space.


Modified Fort Space is not Zero Dimensional Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not zero dimensional.


Equivalence of Definitions of Topologically Equivalent Metrics {{TFAE|def = Topologically Equivalent Metrics}} Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.