Finite Union of Compact Sets is Compact

Let $T = \left({S, \tau}\right)$ be a topological space. Let $n \in \N$ be a natural number. Let $\left\langle{U_i}\right\rangle_{1 \mathop \le i \mathop \le n}$ be a finite sequence of compact subsets of $T$. Let $\mathcal U_n := \displaystyle \bigcup_{i \mathop = 1}^n U_i$ be the union of $\left\langle{U_i}\right\rangle$. Then $\mathcal U_n$ is compact in $T$.


Principle of Mathematical Induction

Let $\map P n$ be a propositional function depending on $n \in \Z$. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad \map P {n_0}$ is true :$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.


Principle of Finite Induction

Let $S \subseteq \Z$ be a subset of the integers. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad n_0 \in S$ :$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$ Then: :$\forall n \ge n_0: n \in S$ That is: :$S = \set {n \in \Z: n \ge n_0}$


Principle of Mathematical Induction/Peano Structure

Let $\struct {P, s, 0}$ be a Peano structure. Let $\map Q n$ be a propositional function depending on $n \in P$. Suppose that: :$(1): \quad \map Q 0$ is true :$(2): \quad \forall n \in P: \map Q n \implies \map Q {\map s n}$ Then: :$\forall n \in P: \map Q n$


Principle of Mathematical Induction/Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $T \subseteq S$ such that $0 \in T$ and $n \in T \implies n \circ 1 \in T$. Then $T = S$.


Consecutive Fibonacci Numbers are Coprime

Let $F_k$ be the $k$th Fibonacci number. Then: :$\forall n \ge 2: \gcd \set {F_n, F_{n + 1} } = 1$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$. That is, a Fibonacci number and the one next to it are coprime.


Divisibility of Fibonacci Number

:$\forall m, n \in \Z_{> 2} : m \divides n \iff F_m \divides F_n$ where $\divides$ denotes divisibility.


Fibonacci Number in terms of Smaller Fibonacci Numbers

:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$


GCD of Fibonacci Numbers

:$\forall m, n \in \Z_{> 2}: \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.


Catalan's Identity

:${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$


Fibonacci Number with Negative Index

:$\forall n \in \Z_{> 0} : F_{-n} = \left({-1}\right)^{n + 1} F_n$


Fibonacci Number in terms of Larger Fibonacci Numbers

:$\forall m, n \in \Z_{>0} : F_{m - n} = \left({-1}\right)^{n + 1} F_{m - 1} F_n + \left({-1}\right)^n F_m F_{n - 1}$


Vajda's Identity

==== Formulation 1 ==== {{:Vajda's Identity/Formulation 1}} ==== Formulation 2 ==== {{:Vajda's Identity/Formulation 2}}


Determinant of Matrix Product/General Case

Let $\mathbf A_1, \mathbf A_2, \cdots, \mathbf A_n$ be square matrices of order $n$, where $n > 1$. Then: :$\map \det {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_n} = \map \det {\mathbf A_1} \map \det {\mathbf A_2} \cdots \map \det {\mathbf A_n}$


Opposite Sides Equal implies Parallelogram

Let $ABCD$ be a convex quadrilateral with $AB = CD$ and $BC = AD$. Then $ABCD$ is a parallelogram.


Properties of Fibonacci Numbers

Let $F_n$ denote the $n$th Fibonacci number: {{:Definition:Fibonacci Number}}


Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $p \in S$. Let $T \subseteq S$ such that: :$x \in T \implies p \preceq x \land \paren {x \in T \implies x \circ 1 \in T}$ Then: :$S \setminus S_p \subseteq T$ where: :$\setminus$ denotes set difference :$S_p$ denotes the set of all elements of $S$ preceding $p$.


Principle of Mathematical Induction for Minimal Infinite Successor Set

Let $\omega$ be the minimal infinite successor set. Let $S \subseteq \omega$. Suppose that: :$(1): \quad \varnothing \in S$ :$(2): \quad \forall x: x \in S \implies x^+ \in S$ where $x^+$ is the successor set of $x$. Then: :$S = \omega$ </onlyinclude>


Principle of Mathematical Induction for Natural Numbers in Real Numbers

Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Suppose that $A \subseteq \N$ is an inductive set. Then $A = \N$.


Chinese Remainder Theorem (Commutative Algebra)

Let $A$ be a commutative ring. {{explain|Presumably a Definition:Commutative and Unitary Ring is actually required here?}} Let $I_1, \ldots, I_k$ for some $k \ge 1$ be pairwise coprime ideals in $A$, that is: : $\forall i \ne j: I_i + I_j = A$ Then there is an isomorphism of rings: :$A / \left({I_1 \cap \ldots \cap I_k}\right) \to A / I_1 \times \cdots \times A / I_k$ which is induced by the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_k$ defined as: :$\phi \left({x}\right) = \left({x + I_1, \ldots, x + I_k}\right)$ which passes through the quotient. {{explain|what is meant by "passes through the quotient"?}}


Sine of i

:$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$


Cosine of i

:$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$


Tangent of i

:$\tan i = \left({\dfrac {e^2 - 1} {e^2 + 1} }\right) i$


Cosecant of i

:$\csc i = \left({\dfrac {2 e} {1 - e^2} }\right) i$


Secant of i

:$\sec i = \dfrac {2 e} {e^2 + 1}$


Cotangent of i

:$\cot i = \left({\dfrac {1 + e^2} {1 - e^2} }\right) i$


Sine in terms of Cosine

{{begin-eqn}} {{eqn | l = \sin x       | r = +\sqrt {1 - \cos ^2 x}       | c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ }} {{eqn | l = \sin x       | r = -\sqrt {1 - \cos ^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ }} {{end-eqn}}


Sine is Reciprocal of Cosecant

:$\sin \theta = \dfrac 1 {\csc \theta}$


Tangent is Reciprocal of Cotangent

:$\tan \theta = \dfrac 1 {\cot \theta}$


Sine of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$ where: :$\sin$ denotes the sine function (real and complex) :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function.


Cosine of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$ where: :$\cos$ denotes the cosine function (real and complex) :$\sin$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function


Tangent of Complex Number

Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:


Sine in terms of Secant

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


Sign of Sine

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


Second Principle of Mathematical Induction

Let $\map P n$ be a propositional function depending on $n \in \Z$. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad \map P {n_0}$ is true :$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \ge n_0$. This process is called '''proof by (mathematical) induction'''.


Second Principle of Finite Induction

Let $S \subseteq \Z$ be a subset of the integers. Let $n_0 \in \Z$ be given. Suppose that: :$(1): \quad n_0 \in S$ :$(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$\forall n \ge n_0: n \in S$


Sign of Cosine

Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \cos x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \cos x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}} where $\cos$ is the real cosine function.


Sign of Tangent

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


Reciprocal of Strictly Negative Real Number is Strictly Negative

:$\forall x \in \R: x < 0 \implies \dfrac 1 x < 0$


Sign of Secant

Let $x$ be a real number. {{begin-eqn}} {{eqn | l = \sec x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \sec x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}} where $\sec$ is the real secant function.


Cosine in terms of Tangent

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


Tangent in terms of Secant

{{begin-eqn}} {{eqn | l = \tan x       | r = +\sqrt {\sec^2 x - 1}       | c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ }} {{eqn | l = \tan x       | r = -\sqrt {\sec^2 x - 1}       | c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ }} {{end-eqn}}


Union of Left-Total Relations is Left-Total

Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be left-total relations. Then $\mathcal R_1 \cup \mathcal R_2$ is left-total.


Union of Inverse is Inverse of Union

Let for $i \in \left\{1,2\right\}$ $\mathcal R_i \subseteq S_i \times T_i$ be a relation on $S_i \times T_i$. Let $\mathcal R_i^{-1} \subseteq T_i \times S_i$ be the inverse of $\mathcal R_i$. Then $\mathcal R_1^{-1} \cup \mathcal R_2^{-1} = \left(\mathcal R_1 \cup \mathcal R_2\right)^{-1}$


Condition for Darboux Integrability

Let $\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\closedint a b$. Then $f$ is Darboux integrable {{iff}}: :for every $\epsilon \in \R_{>0}$, there exists a finite subdivision $S$ of $\closedint a b$ such that $\map U S – \map L S < \epsilon$ where :$\map U S$ is the upper sum of $f$ on $\closedint a b$ with respect to $S$ :$\map L S$ is the lower sum of $f$ on $\closedint a b$ with respect to $S$


Open Set minus Closed Set is Open

Let $T = \left({S, \tau}\right)$ be a topological space. For $A \subseteq S$ denote by $\complement_S \left({A}\right)$ the relative complement of $A$ in $S$. Let $U \in \tau$ and $\complement_S \left({V}\right) \in \tau$. Then: :$U \setminus V \in \tau$ and: :$\complement_S \left({V \setminus U}\right) \in \tau$


Union of Right-Total Relations is Right-Total

Let $S_1, S_2, T_1, T_2$ be sets or classes. Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be right-total relations. Then $\mathcal R_1 \cup \mathcal R_2$ is right-total.


Primitive of x over a x + b squared by p x + q/Corollary

:$\displaystyle \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } + \frac x {a x + b} } + C$


Arctangent of Imaginary Number

Let $x$ belong to the open real interval $\openint {-1} 1$. Then: :$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$ where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit.


Equivalence of Definitions of Real Inverse Hyperbolic Tangent

Let $S$ denote the open real interval: : $S := \left({-1 \,.\,.\, 1}\right)$ {{TFAE|def = Real Inverse Hyperbolic Tangent}}


Square Root is Strictly Increasing

The positive square root function is strictly increasing, that is: :$ \forall x,y \in \R_{>0}: x < y \implies \sqrt x < \sqrt y$


Minimum of Real Hyperbolic Cosine Function

Let $x$ be a real number. Then: :$\cosh x \ge 1$ where $\cosh$ denotes the hyperbolic cosine function.


Exponential of Real Number is Strictly Positive

Let $x$ be a real number. Let $\exp$ denote the (real) exponential function. Then: :$\forall x \in \R : \exp x > 0$


Derivative of Power of Function

Let $\map u x$ be a differentiable real function of $x$. Let $n$ be a real number such that $n \ne -1$. Then: :$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$


Natural Logarithm of e is 1

:$\ln e = 1$


Real Inverse Hyperbolic Cosine is Strictly Increasing

The real inverse hyperbolic cosine function is strictly increasing, that is: :$\forall x, y \ge 1 : x < y \implies \cosh^{-1} x < \cosh^{-1} y$


Laplace Transform of Constant Multiple

Let $a \in \C$ or $\R$ be constant. Then: :$a \laptrans {\map f {a t} } = \map F {\dfrac s a}$


Laplace Transform of Function of t minus a

Let $g$ be the function defined as: :$\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$ Then: :$\laptrans {\map g t} = e^{-a s} \map F s$


Seifert-van Kampen Theorem

The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions.


Functions of Independent Random Variables are Independent

Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively. Then $\map g X$ and $\map h Y$ are independent random variables.


Multiplication Property of Characteristic Functions

Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\phi_X$ and $\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively. Then: :$\phi_{X + Y} = \phi_X \phi_Y$


Relationship between Limit Inferior and Lower Limit

Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function. Let $\sequence {s_n}_{n \mathop \in \N}$ be a convergent sequence in $S$ such that $s_n \to \bar s$. Then the lower limit of $f$ at $\bar s$ is bounded above by the limit inferior of $\sequence {\map f {s_n} }$: :$\displaystyle \liminf_{s \mathop \to \bar s} \map f s \le \liminf_{n \mathop \to \infty} \map f {s_n}$


Sequence on Finite Product Space Converges to Point iff Projections Converge to Projections of Point

Let $N \in \N$. For all $k \in \set {1, \ldots, N}$, let $T_k = \struct {X_k, \tau_k}$ be topological spaces. Let $\displaystyle X = \prod_{k \mathop = 1}^N X_k$ be the cartesian product of $X_1, \ldots, X_N$. Let $\tau$ be the product topology on $X$. Denote by $\pr_k : X \to X_k$ the projection from $X$ onto $X_k$. Let $\sequence {x_n}$ be a sequence on $X$ and let $x \in X$. Then $\sequence {x_n}$ converges to $x$ {{iff}}: :for all $k \in \set {1, \ldots, N}$ the sequence $\sequence {\map {\pr_k} {x_n} }$ converges to $\map {\pr_k} x$.


Real Convergent Sequence is Cauchy Sequence

Every convergent real sequence in $\R$ is a Cauchy sequence.


Real Sequence is Cauchy iff Convergent

Let $\sequence {a_n}$ be a sequence in $\R$. Then $\sequence {a_n}$ is a Cauchy sequence {{iff}} it is convergent.


Complex Sequence is Cauchy iff Convergent

Let $\sequence {z_n}$ be a complex sequence. Then $\sequence {z_n}$ is a Cauchy sequence {{iff}} it is convergent.


Sum of Arctangents

:$\arctan a + \arctan b = \arctan \dfrac {a + b} {1 - a b}$ where $\arctan$ denotes the arctangent.


Difference of Arctangents

:$\arctan a - \arctan b = \arctan \dfrac {a - b} {1 + a b}$ where $\arctan$ denotes the arctangent.


Sum of Arccotangents

:$\arccot a + \arccot b = \arccot \dfrac {a b - 1} {a + b}$ where $\arccot$ denotes the arccotangent.


Difference of Arccotangents

:$\arccot a - \arccot b = \arccot \dfrac {a b + 1} {a - b}$ where $\arccot$ denotes the arccotangent.


Multiple Angle Formula for Tangent

:$\displaystyle \tan \left({n \theta}\right) = \frac {\displaystyle \sum_{i \mathop = 0}^{\left\lfloor{\frac {n - 1} 2}\right\rfloor} \left({-1}\right)^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\displaystyle \sum_{i \mathop = 0}^{\left\lfloor{\frac n 2}\right\rfloor} \left({-1}\right)^i \binom n {2 i} \tan^{2 i}\theta}$


Modulus of Complex Number equals its Distance from Origin

The modulus of a complex number equals its distance from the origin on the complex plane.


Empty Set is Countable

The empty set $\O$ is countable.


Area of Isosceles Triangle in terms of Sides

Let $\triangle ABC$ be an isosceles triangle whose apex is $A$. Let $r$ be the length of a leg of $\triangle ABC$. Let $b$ be the length of the base of $\triangle ABC$. Then the area $\mathcal A$ of $\triangle ABC$ is given by: :$\mathcal A = \dfrac b 4 \sqrt{4 r^2 - b^2}$


Even Function Times Even Function is Even

Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ Let $f, g: X \to \R$ be two even functions. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: X \to \R$ is also an even function.


Odd Function Times Even Function is Odd

Let $X \subset \R$ be a symmetric set of real numbers: :$\forall x \in X: -x \in X$ Let $f: X \to \R$ be an odd function. Let $g: X \to \R$ be an even function. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: X \to \R$ is an odd function.


Odd Function Times Odd Function is Even

Let $S \subset \R$ be a symmetric set of real numbers: :$\forall x \in S: -x \in X$ Let $f, g: X \to \R$ be two odd functions. Let $f \cdot g$ denote the pointwise product of $f$ and $g$. Then $\paren {f \cdot g}: S \to \R$ is an even function.


Mellin Transform of Exponential

Let $a$ be a complex constant and $e^t$ be the complex exponential. Let $\MM$ be the Mellin transform. Then: :$\map {\MM \set {e^{-a t} } } s = a^{-s} \, \map \Gamma s$ where $\map \Re a, \map \Re s > 0$


Mellin Transform of Dirac Delta Function

Let $\delta \left({t}\right)$ be the Dirac delta function. Let $c$ be a positive real number. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ { \delta_c \left({t}\right)} \right\} \left({s}\right) = c^{s-1}$


Mellin Transform of Power Times Function

Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {t^n f \left({t}\right)} \right\} \left({s}\right) = \mathcal M \left\{ {f \left({t}\right)}\right\} \left({s + n}\right)$ given that both transforms exist.


Mellin Transform of Dirac Delta Function by Function

Let $f: \R \to \R$ be a function. Let $c \in \R_{>0}$ be a positive constant real number. Let $\delta_c \left({t}\right)$ be the Dirac delta function. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {\delta_c \left({t}\right) f \left({t}\right)} \right\} \left({s}\right) = c^{s - 1} f \left({c}\right)$


Unity of Ring is Idempotent

Let $\left({R, +, \circ}\right)$ be a ring with unity whose unity is $1_R$. Then $1_R$ is an idempotent element of $R$ under the ring product $\circ$: :$1_R \circ 1_R = 1_R$


Mellin Transform of Higher Order Exponential

Let $a$ be a complex constant. Let $n$ be a natural number. Let $e^t$ be the complex exponential of $t$. Let $\MM$ be the Mellin transform. Then: :$\map {\MM \set {e^{-a t^n} } } s = \dfrac {a^{-s/n} } n \map \Gamma {\dfrac s n}$ where $\map \Gamma z$ is the Gamma function and $\map \Re a$, $\map \Re s > 0$.


Complex Numbers as Quotient Ring of Real Polynomial

Let $\C$ be the set of complex numbers. Let $P \left[{x}\right]$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real. Let $\left\langle{x^2 + 1}\right\rangle = \left\{ {Q \left({x}\right) \left({x^2 + 1}\right): Q \left({x}\right) \in P \left[{x}\right]}\right\}$ be the ideal generated by $x^2 + 1$ in $P \left[{x}\right]$. Let $D = P \left[{x}\right] / \left\langle{x^2 + 1}\right\rangle$ be the quotient of $P \left[{x}\right]$ modulo $\left\langle{x^2 + 1}\right\rangle$. Then: :$\left({\C, +, \times}\right) \cong \left({D, +, \times}\right)$


Quaternion Modulus in Terms of Conjugate

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\left\vert{\mathbf x}\right\vert$ be the modulus of $\mathbf x$. Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$. Then: : $\left\vert{\mathbf x}\right\vert^2 \mathbf 1 = \mathbf x \overline{\mathbf x}$


Sufficient Condition for Quaternion Multiplication to Commute

In general, quaternion multiplication does not commute. But, for $\mathbf x,\mathbf y \in \H$, $\mathbf x \times \mathbf y = \mathbf y \times \mathbf x$ if any one of the following conditions hold: {{begin-eqn}} {{eqn | n = 1a       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + b \mathbf i + 0 \mathbf j + 0 \mathbf k: a, b \in \R} }} {{eqn | n = 1b       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + c \mathbf j + 0 \mathbf k: a, c \in \R} }} {{eqn | n = 1c       | l = \mathbf x, \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + d \mathbf k: a, d \in \R} }} {{eqn | n = 2a       | l = \mathbf x       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k: a \in \R} }} {{eqn | n = 2b       | l = \mathbf y       | o = \in       | r = \set {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k: a \in \R} }} {{eqn | n = 3       | l = \mathbf x       | r = \paren {a \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k} \times \overline {\mathbf y}: a \in \R }} {{end-eqn}}


Complex Conjugation is Involution

Let $z = x + i y$ be a complex number. Let $\overline z$ denote the complex conjugate of $z$. Then the operation of complex conjugation is an involution: :$\overline {\paren {\overline z} } = z$


Quaternion Modulus of Conjugate

Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline z$ be the conjugate of $z$. Let $\cmod z$ be the quaternion modulus of $z$. Then: :$\cmod {\overline z} = \cmod z$


Quaternion Conjugation is Involution

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ denote the quaternion conjugate of $\mathbf x$. Then the operation of quaternion conjugation is an involution: :$\overline {\paren {\overline {\mathbf x} } } = \mathbf x$


Sum of Quaternion Conjugates

Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. Then: :$\overline {\mathbf x + \mathbf y} = \overline {\mathbf x} + \overline {\mathbf y}$


Principle of Recursive Definition for Peano Structure

Let $\struct {P, 0, s}$ be a Peano structure. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Then there exists exactly one mapping $f: P \to T$ such that: :$\forall x \in P: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = \map s n \end{cases}$


Principle of Recursive Definition for Minimal Infinite Successor Set

Let $\omega$ be the minimal infinite successor set. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Then there exists exactly one mapping $f: \omega \to T$ such that: :$\forall x \in \omega: \map f x = \begin {cases} a & : x = \O \\ \map g {\map f n} & : x = n^+ \end {cases}$ where $n^+$ is the successor set of $n$.


Ordering on 1-Based Natural Numbers is Trichotomy

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\N_{>0}$. Then exactly one of the following is true: :$(1): \quad a = b$ :$(2): \quad a > b$ :$(3): \quad a < b$ That is, $<$ is a trichotomy on $\N_{> 0}$.


Product of Quaternion Conjugates

Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions. Let $\overline{\mathbf x}$ be the conjugate of $\mathbf x$. Then: :$\overline{\mathbf x \times \mathbf y} = \overline{\mathbf y} \times \overline{\mathbf x}$ but in general: :$\overline{\mathbf x \times \mathbf y} \ne \overline{\mathbf x} \times \overline{\mathbf y}$


Quaternion Modulus of Product of Quaternions

Let $\mathbf x, \mathbf y$ be quaternions. Let $\left\vert{\mathbf x}\right\vert$ be the modulus of $\mathbf x$. Then: :$\left\vert{\mathbf x \mathbf y}\right\vert = \left\vert{\mathbf x}\right\vert \left\vert{\mathbf y}\right\vert$


Octonion Conjugation is Involution

Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a octonion. Let $\overline x$ be the conjugate of $x$. Then: :$\overline \cdot: x \mapsto \overline x$ is an involution. That is: :$\overline {\paren {\overline x} } = x$


Ordering on 1-Based Natural Numbers is Compatible with Addition

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $+$ denote addition on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, n \in \N_{>0}: a < b \implies a + n < b + n$ That is, $>$ is compatible with $+$ on $\N_{>0}$.


Ordering on 1-Based Natural Numbers is Compatible with Multiplication

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ denote multiplication on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, n \in \N_{>0}: a < b \implies a \times n < b \times n$ That is, $>$ is compatible with $\times$ on $\N_{>0}$.


Integral between Limits is Independent of Direction

Let $f$ be a real function which is integrable on the interval $\openint a b$. Then: :$\ds \int_a^b \map f x \rd x = \int_a^b \map f {a + b - x} \rd x$


Countable Set equals Range of Sequence

Let $S$ be a set. Then $S$ is countable {{iff}} there exists a sequence $\sequence {s_i}_{i \mathop \in N}$ where $N$ is a subset of $\N$ such that $S$ equals the range of $\sequence {s_i}_{i \mathop \in N}$.


Initial Segment of Natural Numbers determined by Zero is Empty

Let $\N_k$ denote the initial segment of the natural numbers determined by $k$: :$\N_k = \set {0, 1, 2, 3, \ldots, k - 1}$ Then $\N_0 = \O$.


Initial Segment of One-Based Natural Numbers determined by Zero is Empty

Let $\N^*_k$ denote the initial segment of the one-based natural numbers determined by $k$: :$\N^*_k = \set {1, 2, 3, \ldots, k - 1, k}$ Then $\N^*_0 = \O$.


Heine-Borel Theorem/Real Line/Closed and Bounded Interval

Let $\left[{a \,.\,.\, b}\right]$, $a < b$, be a closed and bounded real interval. Let $S$ be a set of open real sets. Let $S$ be a cover of $\left[{a \,.\,.\, b}\right]$. Then there is a finite subset of $S$ that covers $\left[{a \,.\,.\, b}\right]$.


Addition on 1-Based Natural Numbers is Cancellable

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $+$ be addition on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a + c = b + c \implies a = b$ :$\forall a, b, c \in \N_{>0}: a + b = a + c \implies b = c$ That is, $+$ is cancellable on $\N_{>0}$.


Addition on 1-Based Natural Numbers is Cancellable for Ordering

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $<$ be the strict ordering on $\N_{>0}$. Let $+$ be addition on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a + c < b + c \implies a < b$ :$\forall a, b, c \in \N_{>0}: a + b < a + c \implies b < c$ That is, $+$ is cancellable on $\N_{>0}$ for $<$.


Multiplication on 1-Based Natural Numbers is Cancellable

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ be multiplication on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a \times c = b \times c \implies a = b$ :$\forall a, b, c \in \N_{>0}: a \times b = a \times c \implies b = c$ That is, $\times$ is cancellable on $\N_{>0}$.


Multiplication on 1-Based Natural Numbers is Cancellable for Ordering

Let $\N_{> 0}$ be the $1$-based natural numbers. Let $\times$ be multiplication on $\N_{>0}$. Let $<$ be the strict ordering on $\N_{>0}$. Then: :$\forall a, b, c \in \N_{>0}: a \times c < b \times c \implies a < b$ :$\forall a, b, c \in \N_{>0}: a \times b < a \times c \implies b < c$ That is, $\times$ is cancellable on $\N_{>0}$ for $<$.


Index of Trivial Subgroup is Cardinality of Group

Let $G$ be a group whose identity element is $e$. Let $\set e$ be the trivial subgroup of $G$. Then: :$\index G {\set e} = \order G$ where: :$\index G {\set e}$ denotes the index of $\set e$ in $G$ :$\order G$ denotes the cardinality of $G$.


Index is One iff Subgroup equals Group

Let $G$ be a group whose identity element is $e$. Let $H$ be a subgroup of $G$. Then: :$\index G H = 1 \iff G = H$ where $\index G H$ denotes the index of $H$ in $G$.


Set Equality is Equivalence Relation

Let $S$ be a set. Set equality is an equivalence relation on the power set $\powerset S$ of $S$.


Subset Relation is Antisymmetric

The relation "is a subset of" is antisymmetric: :$\paren {R \subseteq S} \land \paren {S \subseteq R} \iff R = S$


There Exists No Universal Set

There exists no set which is an absolutely universal set. That is: :$\neg \left({\exists \mathcal U: \forall T: T \in \mathcal U}\right)$ where $T$ is any arbitrary object at all. That is, a set that contains ''everything'' cannot exist.


Union of Disjoint Singletons is Doubleton

Let $\set a$ and $\set b$ be singletons such that $a \ne b$. Then: :$\set a \cup \set b = \set {a, b}$


Distributive Laws

== Intersection Distributes over Union == {{:Intersection Distributes over Union}}


Intersection of Empty Set/Paradoxical Implications

Although it appears counter-intuitive, the reasoning is sound. This result is therefore classed as a veridical paradox. {{AuthorRef|Paul R. Halmos}} declares, in Section $5$ of his {{BookLink|Naive Set Theory|Paul R. Halmos}} of $1960$ that: :''There is no profound problem here; it is merely a nuisance to be forced always to be making qualifications and exceptions just because some set somewhere along some construction might turn out to be empty. There is nothing to be done about this; it is just a fact of life.'' However, later in that same work (in Section $9$, in the context of indexed families of sets) he says that: :''... an empty intersection does not make sense.'' a sentiment which is repeated in the $2008$ collaboration with {{AuthorRef|Steven Givant}}, {{BookLink|Introduction to Boolean Algebras|Paul Halmos}}, p. $457$. In {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}, the author recognizes the result, but does not adopt it: :''If one has a given large set $X$ that is specified at the outset of the discussion to be one's "universe of discourse," and one considers only subsets of $X$ throughout, it is reasonable to let $\displaystyle \bigcap_{A \mathop \in \AA} A = X$ when $\AA$ is empty. Not all mathematicians follow this convention, however. To avoid difficulty, we shall not define the intersection when $\AA$ is empty.'' {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty}} accepts this result, but cautiously: :''A natural mnemonic for these extreme cases is that $\bigcap \SS$ "grows larger" as $\SS$ "grows smaller", and $\bigcup \SS$ grows smaller as $\SS$ grows smaller. No other convention is possible, but the case $\SS = \O$ will often be treated redundantly by itself in definitions and proofs, as a reminder of the null case.'' {{BookReference|Set Theory|1999|András Hajnal|author2 = Peter Hamburger}} dismiss it casually: :''As usual, we adopt the convention that in case $A = \O$ the expression $\bigcap A$ is defined only in case we work with the subsets of an underlying set $X$. In this case we put $\bigcap A = X$.'' and {{BookReference|Programming, Games and Transportation Networks|1965|Claude Berge|author2 = A. Ghouila-Houri}} do not even acknowledge that there may be a problem in the first place: :''In the case where $I = \O$, we have ::$\displaystyle \bigcap_{i \mathop \in I} A_i = X$;'' :''this moreover is the only case where $X$ plays a role; in fact, if $I$ is not empty, clearly we have: ::$\displaystyle \bigcap_{i \mathop \in I} A_i = \set {x \mid x \in A_i \text { for every } i \in I}$.


Ordered Pair/Kuratowski Formalization/Motivation

The only reason for the Kuratowski formalization of ordered pairs: :$\tuple {a, b} = \set {\set a, \set {a, b} }$ is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory (ZF). Once that has been demonstrated, there is no need to invoke it again. The fact that this formulation allows that: :$\tuple {a, b} = \tuple {c, d} \iff a = c, b = d$ is its stated aim. The fact that $\set {a, b} \in \tuple {a, b}$ is an unfortunate side-effect brought about by means of the definition. It would be possible to add another axiom to ZF or ZFC specifically to allow for ordered pairs to be defined, and in some systems of axiomatic set theory this is what is done.


Heine-Borel Theorem/Real Line/Closed and Bounded Set

Let $F$ be a closed and bounded real set. Let $C$ be a set of open real sets. Let $C$ be a cover of $F$. Then there is a finite subset of $C$ that covers $F$.


Cardinality of Set of All Mappings from Empty Set

Let $T$ be a set. Let $T^\O$ denote the set of all mappings from $\O$ to $S$. Then: :$\card {T^\O} = 1$ where $\card {T^\O}$ denotes the cardinality of $\O^S$.


Cardinality of Set of All Mappings to Empty Set

Let $S$ be a set. Let $\O^S$ be the set of all mappings from $S$ to $\O$. Then: :$\card {\O^S} = \begin{cases} 1 & : S = \O \\  0 & : S \ne \O \end{cases}$ where $\card {\O^S}$ denotes the cardinality of $\O^S$.


Intersection Distributes over Union/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: :$\displaystyle \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$ where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Cartesian Product of Unions/General Result

Let $I$ and $J$ be indexing sets. Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively. Then: :$\displaystyle \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j} = \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren {A_i \times B_j}$ where: :$\displaystyle \bigcup_{i \mathop \in I} A_i$ denotes the union of $\family {A_i}_{i \mathop \in I}$ and so on :$\times$ denotes Cartesian product.


Preimages All Exist iff Surjection/Corollary

:$\forall B \subseteq T, B \ne \O: f^{-1} \sqbrk B \ne \O$ {{iff}}: :$f$ is a surjection where $f^{-1} \sqbrk B$ denotes the preimage of $B \subseteq T$.


Image of Preimage under Relation is Subset

Let $\mathcal R \subseteq S \times T$ be a relation. Then: :$B \subseteq T \implies \paren {\mathcal R \circ \mathcal R^{-1} } \sqbrk B \subseteq B$ where: :$\mathcal R \sqbrk B$ denotes the image of $B$ under $\mathcal R$ :$\mathcal R^{-1} \sqbrk B$ denotes the preimage of $B$ under $\mathcal R$ :$\mathcal R \circ \mathcal R^{-1}$ denotes composition of $\mathcal R$ and $\mathcal R^{-1}$.


Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping

Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation. Let $\mathrel R^\to$ be the direct image mapping of $\mathcal R$. Let $\mathrel R^\gets$ be the inverse image mapping of $\mathcal R$. Then it is not necessarily the case that: :$\paren {\mathrel R^\to}^{-1} = \mathcal R^\gets$ where $\paren {\mathrel R^\to}^{-1}$ denote the inverse of $\mathrel R^\to$. That is, the inverse of the direct image mapping of $\mathcal R$ does not always equal the inverse image mapping of $\mathcal R$.


Singleton of Power Set less Empty Set is Minimal Subset

Let $S$ be a set which is non-empty. Let $\mathcal C = \mathcal P \left({S}\right) \setminus \varnothing$, that is, the power set of $S$ without the empty set. Let $x \in S$. Then $\left\{{x}\right\}$ is a minimal element of the ordered structure $\left({\mathcal C, \subseteq}\right)$.


Natural Numbers under Multiplication form Semigroup

Let $\N$ be the set of natural numbers. Let $\times$ denote the operation of multiplication on $\N$. The structure $\struct {\N, \times}$ forms a semigroup.


Bernoulli's Hanging Chain Problem

Consider a uniform chain $C$ whose physical properties are as follows: :$C$ is of length $l$ :The mass per unit length of $C$ is $m$ :$C$ is of zero stiffness. Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move. Let $C$ be slightly disturbed in a vertical plane from its position of stable equilibrium. Let $\map y t$ be the horizontal displacement at time $t$ from its position of stable equilibrium of a particle of $C$ which is a vertical distance $x$ from its point of attachment. The $2$nd order ODE describing the motion of $y$ is: :$\dfrac {\d^2 y} {\d t^2} = g \paren {l - x} \dfrac {\d^2 y} {\d x^2} - g \dfrac {\d y} {\d x}$


Real Line Continuity by Inverse of Mapping

Let $f$ be a real function. Let the domain of $f$ be open. Let $f^{-1}$ be the inverse of $f$. Then $f$ is continuous {{iff}}: :for every open real set $O$ that overlaps with the image of $f$, the preimage $f^{-1} \left [{O}\right]$ is open.


Trisecting the Angle/Neusis Construction

Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a neusis construction.


Squaring the Circle by Compass and Straightedge Construction is Impossible

There is no compass and straightedge construction to allow a square to be constructed whose area is equal to that of a given circle.


Difference between Distances from Point on Hyperbola to Foci is Constant

Let $K$ be a hyperbola. Let $F_1$ and $F_2$ be the foci of $K$. Let $P$ be an arbitrary point on $K$. Then the distance from $P$ to $F_1$ minus the distance from $P$ to $F_2$ is constant for all $P$ on $K$.


Time Taken for Body to Fall at Earth's Surface

Let an object $m$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $m$ fall a distance $s$ in time $t$. Then: :$s = \dfrac 1 2 g t^2$ or: :$t = \sqrt {\dfrac {2 s} g}$ where $g$ is the Acceleration Due to Gravity at the height through which $m$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout.


Length of Chord of Circle

Let $C$ be a circle of radius $r$. Let $AB$ be a chord which joins the endpoints of the arc $ADB$. Then: :$AB = 2 r \sin \dfrac \theta 2$ where $\theta$ is the angle subtended by $AB$ at the center of $C$.


Spherical Triangles with Same Angles are Congruent

Two triangles on the surface of a given sphere which have the same angles are congruent.


Sum of Angles of Spherical Triangle

The sum of the angles of a spherical triangle is between $\pi$ and $3 \pi$ radians.


Ptolemy's Theorem

Let $ABCD$ be a cyclic quadrilateral. Then: :$AB \times CD + AD \times BC = AC \times BD$


Spherical Law of Sines

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$


Spherical Law of Cosines

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\cos a = \cos b \cos c + \sin b \sin c \cos A$


Spherical Law of Cosines/Angles

:$\cos A = -\cos B \cos C + \sin B \sin C \cos a$


Spherical Law of Tangents

:$\dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} } = \dfrac {\tan \frac 1 2 \paren {a + b} } {\tan \frac 1 2 \paren {a - b} }$


Cosine of Half Angle for Spherical Triangles

:$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$ where $s = \dfrac {a + b + c} 2$.


Cosine of Half Side for Spherical Triangles

:$\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$ where $S = \dfrac {A + B + C} 2$.


Napier's Rules for Right Angled Spherical Triangles

:410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let either angle $\angle C$ or side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as above: :for $\angle C$ a right angle, the '''interior''' :for $c$ a right angle, the '''exterior''' where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''.


Equation of Circle/Cartesian

:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$


Equation of Circle/Parametric

:$x = a + R \cos t, \ y = b + R \sin t$


Equation of Circle/Cartesian/Corollary 1

The equation: :$A \paren {x^2 + y^2} + B x + C y + D = 0$ is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where: :$R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$ :$\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$ provided: :$A > 0$ :$B^2 + C^2 \ge 4 A D$


Equation of Circle/Cartesian/Corollary 2

The equation of a circle with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 = R^2$


Distance Formula/3 Dimensions

The distance $d$ between two points $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$ in a Cartesian space of 3 dimensions is: :$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2 + \paren {z_1 - z_2}^2}$


Equation of Sphere/Rectangular Coordinates

:$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$


Equation of Sphere/Rectangular Coordinates/Corollary

The equation of a sphere with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 + z^2 = R^2$


Equation of Conic Section

The general conic section can be expressed in Cartesian coordinates in the form: :$a x^2 + b x y + c y^2 + d x + e y + f = 0$ for some $a, b, c, d, e, f \in \R$.


Graph of Quadratic describes Parabola

The locus of the equation defining a quadratic: :$y = a x^2 + b x + c$ describes a parabola.


Graph of Quadratic describes Parabola/Corollary 1

The locus of the equation of the square function: :$y = x^2$ describes a parabola.


Natural Numbers under Addition form Commutative Semigroup

The algebraic structure $\left({\N, +}\right)$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative semigroup.


Gödel's Incompleteness Theorems/First/Corollary

If $T$ is both consistent and complete, it does not contain minimal arithmetic.


Monomorphism that is Split Epimorphism is Split Monomorphism

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism in $\mathbf C$ such that $f$ is a monomorphism and a split epimorphism. Then $f: C \to D$ is a split monomorphism. {{explain|What are $C$ and $D$?}}


Epimorphism that is Split Monomorphism is Split Epimorphism

Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a epimorphism and a split monomorphism. Then $f: C \to D$ is a split epimorphism.


Westwood's Puzzle/Proof 1

:500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$.


Westwood's Puzzle/Proof 2


Vector Cross Product Operator is Bilinear

Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf u       | r = u_i \mathbf i + u_j \mathbf j + u_k \mathbf k }} {{eqn | l = \mathbf v       | r = v_i \mathbf i + v_j \mathbf j + v_k \mathbf k }} {{eqn | l = \mathbf w       | r = w_i \mathbf i + w_j \mathbf j + w_k \mathbf k }} {{end-eqn}} where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$. Let $c$ be a real number. Then: : $\left({c \mathbf u + \mathbf v}\right) \times \mathbf w = c \left({ \mathbf u \times \mathbf w}\right) + \mathbf v \times \mathbf w$


Natural Numbers have No Proper Zero Divisors

Let $\N$ be the natural numbers. Then for all $m, n \in \N$: :$m \times n = 0 \iff m = 0 \lor n = 0$ That is, $\N$ has no proper zero divisors.


Diagonals of Rhombus Bisect Angles

Let $OABC$ be a rhombus. Then: :$(1): \quad OB$ bisects $\angle AOC$ and $\angle ABC$ :$(2): \quad AC$ bisects $\angle OAB$ and $\angle OCB$ 400px


Divergent Sequence with Finite Number of Terms Deleted is Divergent

Let $\left({X, d}\right)$ be a metric space. Let $\left \langle {x_k} \right \rangle$ be a sequence in $X$. Let $\left \langle {x_k} \right \rangle$ be divergent. Let a finite number of terms be deleted from $\left \langle {x_k} \right \rangle$. Then the resulting subsequence is divergent.


Tensor Product is Module

Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. Then: :$T = \displaystyle \bigoplus_{s \mathop \in M \mathop \times N} R s$ is a left module.


Supremum of Subset of Real Numbers is Arbitrarily Close

Let $A \subseteq \R$ be a subset of the real numbers. Let $b$ be a supremum of $A$. Let $\epsilon \in \R_{>0}$. Then: :$\exists x \in A: b − x < \epsilon$


Tensor with Zero Element is Zero in Tensor

Let $R$ be a ring. Let $M$ be a right $R$-module. Let $N$ be a left $R$-module. Let $M \otimes_R N$ denote their tensor product. Then: :$0\otimes_R n = m \otimes_R 0 = 0 \otimes_R 0$ is the zero in $M \otimes_R N$.


Primitive of Arcsecant of x over a/Formulation 1

:$\displaystyle \int \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle x \operatorname{arcsec} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle x \operatorname{arcsec} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$


Cross Product of Vector with Itself is Zero

Let $\mathbf x$ be a vector in a vector space of $3$ dimensions: : $\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$ Then: :$\mathbf x \times \mathbf x = \mathbf 0$ where $\times$ denotes vector cross product.


Reciprocal of Riemann Zeta Function

For $\Re \left({z}\right) > 1$: :$\displaystyle \frac 1 {\zeta \left({z}\right)} = \sum_{k \mathop = 1}^\infty \frac{\mu \left({k}\right)} {k^z}$ where: : $\zeta$ is the Riemann zeta function : $\mu$ is the Möbius function.


Condition for Agreement of Family of Mappings

Let $\left({A_i}\right)_{i \mathop \in I}, \left({B_i}\right)_{i \mathop \in I}$ be families of non empty sets. Let $\left({f_i}\right)_{i \mathop \in I}$ be a family of mappings such that: :$\forall i \in I: f_i \in \mathcal F \left({A_i, B_i}\right)$ {{explain|Clarify: what is $\mathcal F \left({A_i, B_i}\right)$? From the context it can be understood as being the set of all mappings $f_i: A_i \to B_i$ but this is just a guess. The domain and range of each of the $f_i$ could help with being explicitly defined.<br/>Note also that I have taken the liberty of exchanging the difficult-to-read $\mathscr F$ with the more eye-friendly $\mathcal F$.}} We have that: :$\displaystyle \bigcup_{i \mathop \in I} f_i \in \mathcal F \left({\bigcup_{i \mathop \in I} A_i, \bigcup_{i \mathop \in I} B_i }\right)$ {{iff}}: :$\displaystyle \forall i, j \in I: \operatorname{Dom} f_i \cap \operatorname{Dom} f_j \ne \varnothing \implies \left({\forall a \in \left({\operatorname{Dom} f_i \cap \operatorname{Dom} f_j}\right), \left({a, b}\right) \in f_i \implies \left({a, b}\right) \in f_j}\right)$


Supremum of Set of Real Numbers is at least Supremum of Subset

Let $S$ be a set of real numbers. Let $S$ have a supremum. Let $T$ be a non-empty subset of $S$. Then $\sup T$ exists and: :$\sup T \le \sup S$


Supremum of Subset of Union Equals Supremum of Union

Let $S$ be a non-empty real set. Let $S$ have a supremum. Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of non-empty subsets of $S$. Let $\bigcup S_i = S$. Then there exists a $j$ in $\set {1, 2, \ldots, n}$ such that: :$\sup S_j = \sup S$


Condition for Ideal to be Total Ring

Let $\left({A, +, \circ}\right)$ be a commutative ring with unity. Let $I$ be an ideal of $A$ such that the quotient ring $A / I$ is a field. Let $J$ be an ideal of $A$ such that $I \subsetneq J$. Then: :$A = J$


Area between Two Non-Intersecting Chords

Let $AB$ and $CD$ be two chords of a circle whose center is at $O$ and whose radius is $r$. :400px :400px Let $\alpha$ and $\theta$ be respectively the measures in radians of the angles $\angle COD$ and $\angle AOB$. Then the area $\mathcal A$ between the two chords is given by: : $\mathcal A = \dfrac {r^2} 2 \left({\theta - \sin \theta - \alpha + \sin \alpha}\right)$ if $O$ is not included in the area, and: : $\mathcal A = r^2 \left({\pi - \dfrac 1 2 \left({\theta - \sin \theta + \alpha - \sin \alpha}\right)}\right)$ if $O$ is included in the area.


Supremum of Absolute Value of Difference equals Supremum of Difference

Let $S$ be a non-empty real set. Let $\displaystyle \sup_{x, y \mathop \in S} \paren {x - y}$ exist. Then $\displaystyle \sup_{x, y \mathop \in S} \size {x - y}$ exists and: :$\displaystyle \sup_{x, y \mathop \in S} \size {x - y} = \sup_{x, y \mathop \in S} \paren {x - y}$


Supremum of Sum equals Sum of Suprema

Let $A$ and $B$ be non-empty sets of real numbers. Let $A + B$ be $\set {x + y: x \in A, y \in B}$. Let either $A$ and $B$ have suprema or $A + B$ have a supremum. Then all $\sup A$, $\sup B$, and $\sup \paren {A + B}$ exist and: :$\sup \paren {A + B} = \sup A + \sup B$


Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees

Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$. Then $e$ is a bridge of minimum weight in $G$ {{iff}} $e$ belongs to every minimum spanning tree of $G$.


Maximum Weight Edge in all Minimum Spanning Trees is Bridge

Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$ that belongs to every minimum spanning tree of $G$. Let $e$ have maximum weight in $G$. Then $e$ is a bridge in $G$.


Scaling Property of Dirac Delta Function

Let $\delta \left({t}\right)$ be the Dirac delta function. Let $a$ be a non zero constant real number. Then: :$\delta \left({a t}\right) = \dfrac {\delta \left({t}\right)} {\left \vert a \right \vert}$


Supremum of Function is less than Supremum of Greater Function

Let $f$ and $g$ be real functions. Let $S$ be a subset of $\operatorname{Dom} \left({f}\right) \cap \operatorname{Dom} \left({g}\right)$. Let $f \left({x}\right) \le g \left({x}\right)$ for every $x \in S$. Let $\displaystyle \sup_{x \mathop \in S} g \left({x}\right)$ exist. Then $\displaystyle \sup_{x \mathop \in S} f \left({x}\right)$ exists and: :$\displaystyle \sup_{x \mathop \in S} f \left({x}\right) \le \sup_{x \mathop \in S} g \left({x}\right)$.


Characterization of Boundary by Open Sets

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \in \operatorname{Fr} A$ {{iff}}: :for every open set $U$ of $T$: ::if $x \in U$ ::then $A \cap U \ne \varnothing$ and $\complement_S \left({A}\right) \cap U \ne \varnothing$ where: :$\complement_S \left({A}\right) = S \setminus A$ denotes the complement of $A$ in $S$ :$\operatorname{Fr} A$ denotes the boundary of $A$.


Characterization of Closure by Open Sets

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Let $x$ be a point of $T$. Let $A^-$ denote the closure of $A$. Then $x \in A^-$ {{iff}}: :for every open set $U$ of $T$: ::$x \in U \implies A \cap U \ne \varnothing$


Summation is Linear/Sum of Summations

:$\displaystyle \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$


Summation is Linear/Scaling of Summations

:$\displaystyle \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$


Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum

Let $f$ be a real function. Let $S$ be a subset of the domain of $f$. Let $\displaystyle \sup_{x \mathop \in S} \set {\map f x}$ and $\displaystyle \inf_{x \mathop \in S} \set {\map f x}$ exist. Then $\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} }$ exists and: :$\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} } = \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}$


Diaconescu-Goodman-Myhill Theorem

The axiom of choice implies the law of excluded middle.


Characterization of Boundary by Basis

Let $T = \struct {S, \tau}$ be a topological space. Let $\BB \subseteq \tau$ be a basis. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \in \partial A$ {{iff}}: :for every $U \in \BB$: ::if $x \in U$ ::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$ where: :$\relcomp S A = S \setminus A$ denotes the complement of $A$ in $S$ :$\partial A$ denotes the boundary of $A$ in $T$.


Union of Interiors and Boundary Equals Whole Space

Let $T = \struct {S, \tau}$ be a topological space. Let $A$ be a subset of $T$. Then: :$S = \Int A \cup \partial A \cup \Int {A'}$ where: :$A' = S \setminus A$ denotes the complement of $A$ relative to $S$ :$\Int A$ denotes the interior of $A$ :$\partial A$ denotes the boundary of $A$.


Characterization of Derivative by Open Sets

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then :$x \in A'$ {{iff}}: :for every open set $U$ of $T$: ::if $x \in U$ ::then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$ where :$A'$ denotes the derivative of $A$.


Derivative is Included in Closure

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then :$A' \subseteq A^-$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.


Closure Equals Union with Derivative

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^- = A \cup A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.


Derivative of Subset is Subset of Derivative

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$A \subseteq B \implies A' \subseteq B'$ where $A'$ denotes the derivative of $A$ in $T$.


Derivative of Union is Union of Derivatives

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$\left({A \cup B}\right)' = A' \cup B\,'$ where :$A'$ denotes the derivative of $A$.


Derivative of Derivative is Subset of Derivative in T1 Space

Let $T = \left({S, \tau}\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'' \subseteq A'$ where :$A'$ denotes the derivative of $A$


Closure of Derivative is Derivative in T1 Space

Let $T = \left({S, \tau}\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'^- = A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.


Union of Derivatives is Subset of Derivative of Union

Let $T = \struct {S, \tau}$ be a topological space. Let: :$\FF \subseteq \powerset S$ be a set of subsets of $S$ where $\powerset S$ denotes the power set of $S$. Then: :$\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup_{A \mathop \in \FF} A}'$ where $A'$ denotes the derivative of $A$.


Point is Isolated iff not Accumulation Point

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Let $x \in H$. Then: :$x$ is an isolated point in $H$ {{iff}}: :$x$ is not an accumulation point of $H$


Dense-in-itself iff Subset of Derivative

Let $T$ be a topological space. Let $A \subseteq T$. Then: :$A$ is dense-in-itself {{iff}}: :$A \subseteq A'$ where :$A'$ denotes the derivative of $A$.


Closure of Dense-in-itself is Dense-in-itself in T1 Space

Let $T$ be a $T_1$ topological space. Let $A \subseteq T$.  Let $A$ be dense-in-itself. Then the closure $A^-$ of $A$ is also dense-in-itself.


Equivalence of Definitions of Weight of Topological Space

Let $T$ be a topological space. Let $\mathbb B$ be the set of all bases of $T$. The following definitions of the weight of $T$ are equivalent:


Difference of Two Powers/Examples/Difference of Two Cubes

:$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$


Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal

Let $T$ be a topological space. Let $\mathcal F$ be a set of open sets of $T$. There exists a subset $\mathcal G \subseteq \mathcal F$ such that: :$\displaystyle \bigcup \mathcal G = \bigcup \mathcal F$ and: :$\left\vert{\mathcal G}\right\vert \leq w \left({T}\right)$ where: :$w \left({T}\right)$ denotes the weight of $T$ :$\left\vert{\mathcal G}\right\vert$ denotes the cardinality of $\mathcal G$.


Set of Singletons is Smallest Basis of Discrete Space

Let $T = \left({S, \tau}\right)$ be a discrete topological space. Let $\mathcal B = \left\{{\left\{{x}\right\} : x \in S}\right\}$. Then $\mathcal B$ is the smallest basis of $T$. That is: :$\mathcal B$ is a basis of $T$ and: :for every basis $\mathcal C$ of $T$, $\mathcal B \subseteq \mathcal C$.


Cardinality of Set of Singletons

Let $S$ be a set. Let $T = \left\{ {\left\{{x}\right\}: x \in S}\right\}$ be the set of all singletons of elements of $S$. Then: :$\left\vert T \right\vert = \left\vert S \right\vert$ where $\left\vert S \right\vert$ denotes the cardinality of $S$.


Basis has Subset Basis of Cardinality equal to Weight of Space

Let $T = \struct {X, \tau}$ be a topological space. Let $\BB$ be a basis of $T$. Then there exists a basis $\BB_0$ of $T$ such that :$\BB_0 \subseteq \BB$ and $\card {\BB_0} = \map w T$ where: :$\card {\BB_0}$ denotes the cardinality of $\BB_0$ :$\map w T$ denotes the weight of $T$.


Cardinality of Union not greater than Product

Let $\FF$ be a set of sets. Let: :$\size \FF \le \mathbf m$ where  :$\size \FF$ denotes the cardinality of $\FF$ :$\mathbf m$ is cardinal number (possibly infinite). Let: :$\forall A \in \FF: \size A \le \mathbf n$ where :$\mathbf n$ is cardinal number (possibly infinite). Then: :$\ds \size {\bigcup \FF} \le \size {\mathbf m \times \mathbf n} = \mathbf m \mathbf n$


Image of Mapping of Intersections is Smallest Basis

Let $T = \left({X, \tau}\right)$ be a topological space. Let $f:X \to \tau$ be a mapping such that :$\forall x \in X: \left({x \in f \left({x}\right) \land \forall U \in \tau: x \in U \implies f \left({x}\right) \subseteq U}\right)$. Then the image $\operatorname{Im} \left({f}\right)$ is subset of every basis of $T$.


Cardinality of Image of Mapping of Intersections is not greater than Weight of Space

Let $T = \struct {X, \tau}$ be a topological space. Let $f: X \to \tau$ be a mapping such that: :$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$ Then the cardinality of the image of $f$ is no greater than the weight of $T$: $\card {\Img f} \le \map w T$.


Finite Weight Space has Basis equal to Image of Mapping of Intersections

Let $T = \struct {X, \tau}$ be a topological space with finite weight. Then there exist a basis $\BB$ of $T$ and a mapping $f:X \to \tau$ such that: :$\BB = \Img f$ and :$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$ where $\Img f$ denotes the image of $f$.


Rubik's Cube has 54 Facets

Let $S$ be the set of facets of Rubik's cube. Then the cardinality of $S$ is given by: :$\card S = 54$ That is: :A Rubik's cube has $54$ facets.


Equivalence of Definitions of Symmetric Difference/(3) iff (5)

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


Equivalence of Definitions of Symmetric Difference/(2) iff (5)


Even and Odd Integers form Partition of Integers

The odd integers and even integers form a partition of the integers.


Analog between Logic and Set Theory

The concepts of set theory have directly corresponding concepts in logic: :{| border = "1" |- ! style="padding: 2px 10px" | Set Theory ! style="padding: 2px 10px" | Logic |- | align="left" style="padding: 2px 10px"| Set: $S, T$ | align="left" style="padding: 2px 10px"| Statement: $p, q$ |- | align="left" style="padding: 2px 10px"| Union: $S \cup T$ | align="left" style="padding: 2px 10px"| Disjunction: $p \lor q$ |- | align="left" style="padding: 2px 10px"| Intersection: $S \cap T$ | align="left" style="padding: 2px 10px"| Conjunction: $p \land q$ |- | align="left" style="padding: 2px 10px"| Subset: $S \subseteq T$ | align="left" style="padding: 2px 10px"| Conditional: $p \implies q$ |- | align="left" style="padding: 2px 10px"| Symmetric Difference: $S * T$ | align="left" style="padding: 2px 10px"| Exclusive Or: $p \oplus q$ |- | align="left" style="padding: 2px 10px"| Complement: $\relcomp {} S$ | align="left" style="padding: 2px 10px"| Logical Not: $\lnot p$ |- | align="left" style="padding: 2px 10px"| Set Equality: $S = T$ | align="left" style="padding: 2px 10px"| Biconditional: $p \iff q$ |- | align="left" style="padding: 2px 10px"| Venn Diagram | align="left" style="padding: 2px 10px"| Truth Table |}


Cardinality of Set is Finite iff Set is Finite

Let $A$ be a set. $\left\vert{A}\right\vert$ is finite {{iff}} $A$ is finite where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.


Topology Generated by Closed Sets

Let $X$ be a set. Let $\mathcal F$ be a set of subsets of $X$. Suppose that :$\varnothing \in \mathcal F$ and :for every subsets $A$ and $B$ of $X$ if $A, B \in \mathcal F$, then $A \cup B \in \mathcal F$ and :for every subset $\mathcal G \subseteq \mathcal F$, $\bigcap \mathcal G \in \mathcal F$ and :$\tau = \left\{{\complement_X \left(A\right): A \in \mathcal F}\right\}$. Then: :$T = \left( {X, \tau} \right)$ is topological space and :for every subset $A$ of $X$, $A$ is closed in $T$ {{iff}} $A \in \mathcal F$.


Equivalence of Definitions of Countably Infinite Set

Let $S$ be a set. {{TFAE|def = Countably Infinite Set}}


Set of Odd Integers is Countably Infinite

Let $\Bbb O$ be the set of odd integers. Then $\Bbb O$ is countably infinite.


Unique Readability for Polish Notation

Let $\AA$ be an alphabet. Then Polish notation for $\AA$ has the unique readability property.


Product of Rook Matrices is Rook Matrix

Let $\mathbf A$ and $\mathbf B$ be rook matrices. Their product $\mathbf {A B}$ is also a rook matrix.


Topology Defined by Basis

Let $S$ be a set. Let $\mathcal B$ be a set of subsets of $S$. Suppose that :$(B1): \quad \forall A_1, A_2 \in \mathcal B: \forall x \in A_1 \cap A_2: \exists A \in \mathcal B: x \in A \subseteq A_1 \cap A_2$ :$(B2): \quad \forall x \in X: \exists A \in \mathcal B: x \in A$ ::$\tau = \left\{{\bigcup \mathcal G: \mathcal G \subseteq \mathcal B}\right\}$ Then: :$T = \left({S, \tau}\right)$ is a topological space :$\mathcal B$ is a basis of $T$.


Matrix is Non-Invertible iff Product with Non-Zero Vector is Zero

Let $\mathbf A$ be a square matrix of order $n$. Then $\mathbf A$ is non-invertible if there exists a vector $\mathbf v$ of $n$ such that: :$\mathbf v \ne \mathbf 0$ :$\mathbf A \mathbf v = \mathbf 0$ where $\mathbf 0$ is the zero vector.


Equivalence of Definitions of Integer Congruence

Let $m \in \Z_{> 0}$. {{TFAE|def = Congruence (Number Theory)/Integers|view = congruence modulo $m$}}


Krull's Theorem

Let $R$ be a non-null ring with unity. Then $R$ has a maximal ideal.


NAND as Disjunction of Negations

: $p \uparrow q \dashv \vdash \neg p \lor \neg q$


Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication

: $\vdash \left({p \lor p}\right) \implies p$


Rule of Addition/Sequent Form/Formulation 1/Form 2/Proof 2

:$q \vdash p \lor q$


Rule of Addition/Sequent Form/Formulation 2/Form 2

:$\vdash q \implies \left({p \lor q}\right)$


Hilbert Proof System Instance 2 is Consistent

Instance 2 of the Hilbert proof systems $\mathscr H_2$ is consistent.


Set of Local Minimum is Countable

Let $X$ be a subset of $\R$. The set: :$\leftset {x \in X: x}$ is local minimum in $\rightset X$ is countable.


Set of Pairwise Disjoint Intervals is Countable

Let $X$ be a subset of $\mathcal P \left({\R}\right)$ such that: :$(1): \quad X$ is pairwise disjoint: ::::$\forall A,B \in X: A \ne B \implies A \cap B = \varnothing$. :$(2): \quad$ every element of $X$ contains an open interval: ::::$\forall A \in X: \exists x, y \in \R: x < y \land \left({x \,.\,.\, y}\right) \subseteq A$. Then $X$ is countable.


Set is Countable if Cardinality equals Cardinality of Countable Set

Let $X, Y$ be sets. Let: : $\left\vert{X}\right\vert = \left\vert{Y}\right\vert$ where $\left\vert{X}\right\vert$ denotes the cardinality of $X$. If $X$ is countable then $Y$ is countable.


{{:Double Negation/Double Negation Introduction/Sequent Form/Formulation 1}}


Double Negation/Double Negation Elimination/Proof Rule

:If we can conclude $\neg \neg \phi$, then we may infer $\phi$.


Double Negation/Double Negation Elimination/Formulation 1/Sequent Form

{{:Double Negation/Double Negation Elimination/Sequent Form/Formulation 1}}


Biconditional Elimination/Also known as

Some sources refer to the Biconditional Elimination as the rule of '''Biconditional-Conditional'''.


Principle of Non-Contradiction/Explanation

The '''Principle of Non-Contradiction''' can be expressed in natural language as follows: :A statement can not be both true and not true at the same time. This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory. Thus it provides a means of eliminating a logical not from a sequent.


Principle of Non-Contradiction/Proof Rule

:If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.


Solutions of Pythagorean Equation/Primitive

The set of all primitive Pythagorean triples is generated by: :$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ where: :$m, n \in \Z_{>0}$ are (strictly) positive integers :$m \perp n$, that is, $m$ and $n$ are coprime :$m$ and $n$ are of opposite parity :$m > n$


Solutions of Pythagorean Equation/General

Let $x, y, z$ be a solution to the Pythagorean equation. Then $x = k x', y = k y', z = k z'$, where: :$\tuple {x', y', z'}$ is a primitive Pythagorean triple :$k \in \Z: k \ge 1$


Goldbach Conjecture implies Goldbach's Marginal Conjecture

Suppose the Goldbach Conjecture holds: :Every even integer greater than $2$ is the sum of two primes. Then Goldbach's Marginal Conjecture follows: :Every integer greater than $5$ can be written as the sum of three primes.


Infinite Number of Chen Primes

There exists an infinite number of Chen primes. That is, there exists an infinite number of pairs: :$\set {p, p + 2}$ where: :$p$ is a prime :$p + 2$ is either a prime or a semiprime.


Congruent Integers are of same Quadratic Character

Let $p$ be an odd prime. Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$. Let $a \equiv b \pmod p$. Then $a$ and $b$ have the same quadratic character.


Weight of Sorgenfrey Line is Continuum

Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $\map w T = \mathfrak c$ where :$\map w T$ denotes the weight of $T$ :$\mathfrak c$ denotes continuum, the cardinality of real numbers.


Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure

Let $\mathcal B$ be a set of subsets of $\R$, the set of all real numbers. Let: :$\left\vert{\mathcal B}\right\vert < \mathfrak c$ where :$\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$ :$\mathfrak c = \left\vert{\R}\right\vert$ denotes continuum. Then: :$\exists x, y \in \R: x < y \land \left[{x \,.\,.\, y}\right) \notin \left\{{\bigcup \mathcal G: \mathcal G \subseteq \mathcal B}\right\}$


Cardinality of Basis of Sorgenfrey Line not greater than Continuum

Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Let :$\BB = \set {\hointr x y: x, y \in \R \land x < y}$ be the basis of $T$. Then $\card \BB \le \mathfrak c$ where :$\card \BB$ denotes the cardinality of $\BB$ :$\mathfrak c = \card \R$ denotes the continuum.


Construction of Regular Prime p-Gon Exists iff p is Fermat Prime

Let $p$ be a prime number. Then there exists a compass and straightedge construction for a regular $p$-gon {{iff}} $p$ is a Fermat prime.


Cardinalities form Inequality implies Difference is Nonempty

Let $X, Y$ be sets. Let :$\left\vert{X}\right\vert < \left\vert{Y}\right\vert$ where $\left\vert{X}\right\vert$ denotes the cardinality of $X$. Then: :$Y \setminus X \ne \varnothing$


Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum

Let $\BB$ be a set of subsets of $\R$. Let: :$\size \BB < \mathfrak c$ where :$\size \BB$ denotes the cardinality of $\BB$ :$\mathfrak c = \size \R$ denotes continuum. Let :$X = \leftset {x \in \R: \exists U \in \set {\bigcup \GG: \GG \subseteq \BB}: x}$ is local minimum in $\rightset U$ Then: :$\size X < \mathfrak c$


Slope of Secant

Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $AB$ be a secant of $f$ where: :$A = \tuple {x, \map f x}$ :$A = \tuple {x + h, \map f {x + h} }$ Then the slope of $AB$ is given by: :$\dfrac {\map f {x + h} - \map f x} h$


Derivative of Curve at Point

Let $f: \R \to \R$ be a real function. Let the graph $G$ of $f$ be depicted on a Cartesian plane. Then the derivative of $f$ at $x = \xi$ is equal to the tangent to $G$ at $x = \xi$.


Derivative of Square Function

Let $f: \R \to \R$ be the square function: :$\forall x \in \R: \map f x = x^2$ Then the derivative of $f$ is given by: :$\map {f'} x = 2 x$


Countable iff Cardinality not greater than Aleph Zero

Let $X$ be set. $X$ is countable {{iff}}: $\left\vert{X}\right\vert \leq \aleph_0$ where: :$\left\vert{X}\right\vert$ denotes the cardinality of $X$ :$\aleph_0 = \left\vert{\N}\right\vert$ by Aleph Zero equals Cardinality of Naturals.


Aleph Zero equals Cardinality of Naturals

$\aleph_0 = \left\vert{\N}\right\vert$ where :$\aleph$ denotes the aleph mapping, :$\left\vert{\N}\right\vert$ denotes the cardinality of $\N$.


Body under Constant Acceleration/Velocity after Time

: $\mathbf v = \mathbf u + \mathbf a t$


Body under Constant Acceleration/Distance after Time

:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$


Body under Constant Acceleration/Velocity after Distance

:$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf a \cdot \mathbf s$


Body in Free Fall moves in Parabolic Path

A body in free fall above the surface of the Earth follows a path approximating to a parabola.


Aleph Zero is less than Continuum

$\aleph_0 < \mathfrak c$ where :$\aleph$ denotes the aleph mapping, :$\mathfrak c$ denotes continuum, the cardinality of real numbers.


Cardinality of Set less than Cardinality of Power Set

Let $X$ be a set. Then: :$\left\vert{X}\right\vert < \left\vert{\mathcal P \left({X}\right)}\right\vert$ where :$\left\vert{X}\right\vert$ denotes the cardinality of $X$, :$\mathcal P \left({X}\right)$ denotes the power set of $X$.


Volume of Solid of Revolution

Let $f: \R \to \R$ be a real function which is integrable on the interval $\closedint a b$. Let the points be defined: :$A = \tuple {a, \map f a}$ :$B = \tuple {b, \map f b}$ :$C = \tuple {b, 0}$ :$D = \tuple {a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by $\set {\map f x: a \le x \le b}$. Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\displaystyle V = \pi \int_a^b \paren {\map f x}^2 \rd x$


Continuum equals Cardinality of Power Set of Naturals

$\mathfrak c = \card {\powerset \N}$ where :$\powerset \N$ denotes the power set of $\N$ :$\card {\powerset \N}$ denotes the cardinality of $\powerset \N$ :$\mathfrak c = \card \R$ denotes the continuum.


Acceleration is Second Derivative of Displacement with respect to Time

The '''acceleration''' $\mathbf a$ of a body $M$ is the second derivative of the displacement $\mathbf s$ of $M$ from a given point of reference with respect to time $t$: :$\mathbf a = \dfrac {\d^2 \mathbf s} {\d t^2}$


Equation of Catenary/Formulation 2

The '''catenary''' is described by the equation: :$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$ where $a$ is a constant. The lowest point of the chain is at $\tuple {0, a}$.


Cardinality of Power Set is Invariant

Let $X, Y$ be sets. Let $\card X = \card Y$ where $\card X$ denotes the cardinality of $X$. Then: :$\card {\powerset X} = \card {\powerset Y}$ where $\powerset X$ denotes the power set of $X$.


Reals are Isomorphic to Dedekind Cuts

Let $\mathscr D$ be set of all Dedekind cuts of the totally ordered set $\struct {\Q, \le}$. Define a mapping $f: \R \to \mathscr D$ as: :$\forall x \in \R: \map f x = \set {y \in \Q: y < x}$ Then $f$ is a bijection.


Slope of Orthogonal Curves

Let $C_1$ and $C_2$ be curves in a cartesian plane. Let $C_1$ and $C_2$ intersect each other at $P$. Let the slope of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$. Then $C_1$ and $C_2$ are orthogonal {{iff}}: :$m_1 = -\dfrac 1 {m_2}$


Orthogonal Trajectories/Concentric Circles

Consider the one-parameter family of curves: :$(1): \quad x^2 + y^2 = c$ Its family of orthogonal trajectories is given by the equation: :$y = c x$


Angle of Tangent to Radius in Polar Coordinates

Let $C$ be a curve embedded in a plane defined by polar coordinates. Let $P$ be the point at $\polar {r, \theta}$. Then the angle $\psi$ made by the tangent to $C$ at $P$ with the radial coordinate is given by: :$\tan \psi = r \dfrac {\d \theta} {\d r}$


Orthogonal Trajectories/Circles Tangent to Y Axis

Consider the one-parameter family of curves: :$(1): \quad x^2 + y^2 = 2 c x$ which describes the loci of circles tangent to the $y$-axis at the origin. Its family of orthogonal trajectories is given by the equation: :$x^2 + y^2 = 2 c y$ which describes the loci of circles tangent to the $x$-axis at the origin. :600px


Orthogonal Trajectories/Parabolas Tangent to X Axis

Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin: :$(1): \quad y = c x^2$ Its family of orthogonal trajectories is given by the equation: :$x^2 + 2 y^2 = c$ :600px


Orthogonal Trajectories/Exponential Functions

Consider the one-parameter family of curves of graphs of the exponential function: :$(1): \quad y = c e^x$ Its family of orthogonal trajectories is given by the equation: :$y^2 = -2 x + c$ :600px


Orthogonal Trajectories/Parabolas with Focus at Origin

Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis: :$(1): \quad y^2 = 4 c \paren {x + c}$ Its family of orthogonal trajectories is given by the equation: :$y^2 = 4 c \paren {x + c}$ :600px


Speed of Body under Free Fall from Height

Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $B$ fall a distance $s$. Then: :$v = \sqrt {2 g s}$ where: :$v$ is the speed of $B$ after having fallen a distance $s$ :$g$ is the Acceleration Due to Gravity at the height through which $B$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout.


Approximate Motion of Pendulum

Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at a small angle $\alpha$ (less than about $10 \degrees$ or $15 \degrees$) from the vertical and then released. Let $T$ be the period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \pi \sqrt {\dfrac a g}$ where $g$ is the Acceleration Due to Gravity.


Space is Separable iff Density not greater than Aleph Zero

Let $T$ be a topological space. Then: :$T$ is separable {{iff}} $d \left({T}\right) \leq \aleph_0$ where :$d \left({T}\right)$ denotes the density of $T$, :$\aleph$ denotes the aleph mapping.


Boundary of Empty Set is Empty

Let $T$ be a topological space. Then:  :$\partial_T \O = \O$ where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$.


Escape Velocity of Projectile fired Upwards

Let $P$ be a planet. Let $P$ have an Acceleration Due to Gravity at its surface of $g$. Let $P$ have a radius of $R$. Then the escape velocity of $P$ is given by: :$V = \sqrt {2 g R}$


Union of Boundaries

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial A \cup \partial B = \partial \left({A \cup B}\right) \cup \partial \left({A \cap B}\right) \cup \left({\partial A \cap \partial B}\right)$ where $\partial A$ denotes the boundary of $A$.


Length of Arch of Sine Function

The length of one arch of the sine function: :$y = \sin x$ is given by: :$L = 2 \sqrt 2 \map E {\dfrac {\sqrt 2} 2}$ where $E$ denotes the incomplete elliptic integral of the second kind.


Length of Lemniscate of Bernoulli

The total length of the lemniscate of Bernoulli given in polar coordinates as: :$r^2 = a^2 \cos 2 \theta$ is given by: {{begin-eqn}} {{eqn | l = L       | r = 4 a \map F {\sqrt 2, \dfrac \pi 4}       | c =  }} {{eqn | r = \dfrac 1 {\sqrt {2 \pi} } \paren {\map \Gamma {\dfrac 1 4} }^2       | c =  }} {{end-eqn}} where $F$ denotes the incomplete elliptic integral of the first kind.


Boundary of Union is Subset of Union of Boundaries

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial \left({A \cup B}\right) \subseteq \partial A \cup \partial B$ where $\partial A$ denotes the boundary of $A$.


Boundary of Intersection is Subset of Union of Boundaries

Let $T = \struct {S, \tau}$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial \paren {A \cap B} \subseteq \partial A \cup \partial B$ where $\partial A$ denotes the boundary of $A$.


Complete Elliptic Integral of the First Kind as Power Series

The '''complete elliptic integral of the first kind''': :$\displaystyle \map K k = \int_0^{\pi / 2} \frac {\rd \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int_0^1 \frac {\rd v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \map K k       | r = \frac \pi 2 \sum_{i \mathop \ge 0} \paren {\prod_{j \mathop = 1}^i \frac {2 j - 1} {2 j} }^2 k^{2 i}       | c =  }} {{eqn | r = \frac \pi 2 \paren {1 + \paren {\frac 1 2}^2 k^2 + \paren {\frac {1 \cdot 3} {2 \cdot 4} }^2 k^4 + \paren {\frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} }^2 k^6 + \cdots}       | c =  }} {{end-eqn}}


Complete Elliptic Integral of the Second Kind as Power Series

The '''complete elliptic integral of the second kind''': :$\displaystyle \map E k = \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \, \rd \phi = \int_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \, \rd v$ can be expressed as the power series: {{begin-eqn}} {{eqn | l = \map E k       | r = \frac \pi 2 \sum_{i \mathop \ge 0} \paren {\prod_{j \mathop = 1}^i \frac {2 j - 1} { 2 j} }^2 \frac {k^{2 i} } {1 - 2 i}       | c =  }} {{eqn | r = \frac \pi 2 \paren {1 - \paren {\frac 1 2}^2 k^2 - \paren {\frac {1 \cdot 3} {2 \cdot 4} }^2 \frac {k^4} 3 - \paren {\frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} }^2 \frac {k^6} 5 - \cdots}       | c =  }} {{end-eqn}}


Set is Countable iff Cardinality not greater Aleph Zero

Let $X$ be a set. Then: :$X$ is countable {{iff}} $\size X \le \aleph_0$ where :$\size X$ denotes the cardinality of $X$, :$\aleph$ denotes the aleph mapping.


Bernoulli's Theorem

Let the probability of the occurrence of an event be $p$. Let $n$ independent trials be made, with $k$ successes. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac k n = p$


Time of Travel down Brachistochrone

Let a wire $AB$ be curved into the shape of a brachistochrone. Let $AB$ be embedded in a constant and uniform gravitational field where Acceleration Due to Gravity is $g$. Let a bead $P$ be released at $A$ to slide down without friction to $B$. Then the time taken for $P$ to slide from $A$ to $B$ is: :$T = \pi \sqrt {\dfrac a g}$ where $a$ is the radius of the generating circle of the cycloid which forms $AB$.


Time of Travel down Brachistochrone/Corollary

Let a bead $P$ be released from anywhere on the wire between $A$ and $B$ to slide down without friction to $B$. Then the time taken for $P$ to slide to $B$ is: :$T = \pi \sqrt{\dfrac a g}$


Union of Set of Singletons

Let $S$ be a set. Let $T = \set {\set x: x \in S}$ be the set of all singletons of elements of $S$. Then: :$\ds \bigcup T = S$ where $\ds \bigcup T$ denotes the union of $T$.


T1 Space is T1/2 Space

Let $T$ be a $T_1$ topological space. Then $T$ is $T_{\frac 1 2}$ space.


Parabolas Inscribed in Shared Tangent Lines

Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane. Let $\map {y_1} x$ be the equation of the tangent line at $\tuple {Q, \map f Q}$ on $f$. Let $\map {y_2} x$ be the equation of the tangent line at $\tuple {-Q, \map f {-Q} }$ on $f$. Then there exists another function $\map g x$ also embedded in the Euclidean Plane defined as: :$\map g x = -A x^2 + B x + C_2$. with: :tangent lines $\map {y_3} x$ being the equation of the tangent line at $\tuple {Q, \map g Q}$ on $g$ and: :$\map {y_4} x$ being the equation of the tangent line at $\tuple {-Q, \map g {-Q} }$ on $g$. so that the tangent lines $y_3$ and $y_4$ inscribe $\map f x$ and the tangent lines $y_1$ and $y_2$ inscribe $\map g x$.


Characterization of T0 Space by Closures of Singletons

Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::$\forall x, y \in S: x \ne y \implies x \notin \left\{{y}\right\}^- \lor y \notin \left\{{x}\right\}^-$ where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$.


Characterization of T0 Space by Distinct Closures of Singletons

Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::$\forall x, y \in S: x \ne y \implies \left\{{x}\right\}^- \ne \left\{{y}\right\}^-$ where $\left\{{y}\right\}^-$ denotes the closure of $\left\{{y}\right\}$.


Characterization of T0 Space by Closed Sets

Let $T = \left({S, \tau}\right)$ be a topological space. Then :$T$ is a $T_0$ space {{iff}} ::for every points $x, y \in S$ if $x \ne y$ then :::there exists a closed subset $F$ of $S$ such that $x \in F$ and $y \notin F$ ::or :::there exists a closed subset $F$ of $S$ such that $x \notin F$ and $y \in F$


Focus of Ellipse from Major and Minor Axis

Let $K$ be an ellipse whose major axis is $2 a$ and whose minor axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$a^2 = b^2 + c^2$


Equidistance of Ellipse equals Major Axis

Let $K$ be an ellipse whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$d_1 + d_2 = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the major axis of $K$.


Equidistance of Hyperbola equals Transverse Axis

Let $K$ be an hyperbola whose foci are $F_1$ and $F_2$. Let $P$ be an arbitrary point on $K$. Let $d$ be the constant distance such that: :$\left\lvert{d_1 - d_2}\right\rvert = d$ where: : $d_1 = P F_1$ : $d_2 = P F_2$ Then $d$ is equal to the transverse axis of $K$.


Equivalence of Definitions of Ellipse

The following definitions of an ellipse are equivalent:


Equation of Hyperbola in Reduced Form

Let $K$ be an hyperbola aligned in a cartesian plane in reduced form. Let: :the transverse axis of $K$ have length $2 a$ :the conjugate axis of $K$ have length $2 b$.


Equation of Hyperbola in Reduced Form/Cartesian Frame

The equation of $K$ is: :$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$


Focus of Hyperbola from Transverse and Conjugate Axis

Let $K$ be a hyperbola whose transverse axis is $2 a$ and whose conjugate axis is $2 b$. Let $c$ be the distance of the foci of $K$ from the center. Then: :$c^2 = a^2 + b^2$


Equation of Hyperbola in Reduced Form/Cartesian Frame/Parametric Form

The equation of $K$ in parametric form is: :$x = a \cosh \theta, y = b \sinh \theta$


Set of Condensation Points is Subset of Derivative

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^0 \subseteq A'$ where :$A^0$ denotes the set of condensation points of $A$ :$A'$ denotes the derivative of $A$


Closure of Set of Condensation Points equals Itself

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$\left({A^0}\right)^- = A^0$ where :$A^0$ denotes the set of condensation points of $A$ :$A^-$ denotes the closure of $A$


First Order ODE/(x + y) dx = (x - y) dy

is a homogeneous differential equation with general solution: :$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$


Set of Condensation Points is Monotone

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$A \subseteq B \implies {A^0} \subseteq B^0$ where :$A^0$ denotes the set of condensation points of $A$


First Order ODE/x^2 y' = 3 (x^2 + y^2) arctan (y over x) + x y

is a homogeneous differential equation with solution: :$y = x \tan C x^3$


First Order ODE/x sine (y over x) y' = y sine (y over x) + x

is a homogeneous differential equation with solution: :$\cos \dfrac y x + \ln C x = 0$


First Order ODE in form y' = f (a x + b y + c)

The first order ODE: :$\dfrac {\mathrm d y} {\mathrm d x} = f \left({a x + b y + c}\right)$ can be solved by substituting: :$z := a x + b y + c$ to obtain: :$\displaystyle x = \int \frac {\mathrm d z} {b f \left({z}\right) + a}$


First Order ODE/y' = sin^2 (x - y + 1)

The first order ODE: :$\dfrac {\d y} {\d x} = \map {\sin^2} {x - y + 1}^2$ has the general solution: :$\map \tan {x - y + 1} = x + C$


First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))

The first order ODE: :$\dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$ such that: :$ a e \ne b d$ can be solved by substituting: :$x := z - h$ :$y := w - k$ where: :$h = \dfrac {c e - b f} {a e - b d}$ :$k = \dfrac {a f - c d} {a e - b d}$ to obtain: :$\dfrac {\d w} {\d z} = \map F {\dfrac {a z + b w} {d z + e w} }$ which can be solved by the technique of Solution to Homogeneous Differential Equation.


First Order ODE/(x + y + 4) over (x - y - 6)

The first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x - y - 6}$ has the general solution: :$\map \arctan {\dfrac {y + 5} {x - 1} } = \ln \sqrt {\paren {x - 1}^2 + \paren {y + 5}^2} + C$


First Order ODE/(x + y + 4) over (x + y - 6)

The first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$ has the general solution: :$y - x = 5 \, \map \ln {x + y - 1} + C$


First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d/Example

=== $\dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$ === {{:First Order ODE/(x + y + 4) over (x + y - 6)}}


First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f))/Example

=== $\dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x - y - 6}$ === {{:First Order ODE/(x + y + 4) over (x - y - 6)}}


First Order ODE/(x + (2 over y)) dy + y dx = 0

is an exact differential equation with solution: :$x y + 2 \ln y = C$


First Order ODE/(y - x^3) dx + (x + y^3) dy = 0

is an exact differential equation with solution: :$4 x y - x^4 + y^4 = C$


First Order ODE/(y + y cosine x y) dx + (x + x cosine x y) dy = 0

is an exact differential equation with solution: :$x y + \sin x y = C$


First Order ODE/(sine x sine y - x e^y) dy = (e^y + cosine x cosine y) dx

is an exact differential equation with solution: :$\sin x \cos y + x e^y = C$


First Order ODE/(2 x y^3 + y cosine x) dx + (3 x^2 y^2 + sine x) dy

is an exact differential equation with solution: :$x^2 y^3 + y \sin x = C$


Heart Curve

The following equations define curves that are more or less heart shaped:


Limit Point of Subset is Limit Point of Set

Let $T = \struct {S, \tau}$ be a topological space. Let $A, B$ be subset of $S$ such that :$A \subseteq B$ Let $x$ be a point of $S$. Then: :if $x$ is limit point of $A$, then $x$ is limit point of $B$.


Integrating Factor for First Order ODE/Preliminary Work

Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. Let $\map \mu x$ be an integrating factor for $(1)$. Let: :$\map P {x, y} := \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$ Then: :$\dfrac 1 \mu = \dfrac {\map P {x, y} } {N \dfrac {\partial \mu} {\partial x} - M  \dfrac {\partial \mu} {\partial y} }$


Integrating Factor for First Order ODE/Function of One Variable

Suppose that: :$g \left({x}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {N \left({x, y}\right)}$ is a function of $x$ only. Then: :$\mu \left({x}\right) = e^{\int g \left({x}\right) \mathrm d x}$ is an integrating factor for $(1)$. Similarly, suppose that: :$h \left({y}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {M \left({x, y}\right)}$ is a function of $y$ only. Then: :$\mu \left({y}\right) = e^{\int -h \left({y}\right) \mathrm d y}$ is an integrating factor for $(1)$.


Integrating Factor for First Order ODE/Function of Sum of Variables

Suppose that: :$g \left({z}\right) = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {N \left({x, y}\right) - M \left({x, y}\right)}$ is a function of $z$, where $z = x + y$. Then: :$\mu \left({x + y}\right) = \mu \left({z}\right) = e^{\int g \left({z}\right) \mathrm d z}$ is an integrating factor for $(1)$.


Integrating Factor for First Order ODE/Function of Product of Variables

:$\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}} {N y - M x}$ is a function of $z$, where $z = x y$. Then: :$\map \mu {x y} = \map \mu z = e^{\int \map g z \d z}$ is an integrating factor for $(1)$.


Set of Condensation Points of Union is Union of Sets of Condensation Points

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\left({A \cup B}\right)^0 = A^0 \cup B^0$


Set of Condensation Points of Union is Union of Sets of Condensation Points/Lemma

Let $x$ be a point of $S$. Then: :if $x$ is condensation point of $A \cup B$, :then $x$ is condensation point of $A$ or $x$ is condensation point of $B$.


Integrating Factor for First Order ODE/Conclusion

Let the first order ordinary differential equation: :$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$ be non-homogeneous and not exact. Let $\map \mu {x, y}$be an integrating factor for $(1)$. If one of these is the case: :$\mu$ is a function of $x$ only :$\mu$ is a function of $y$ only :$\mu$ is a function of $x + y$ :$\mu$ is a function of $x y$ then: :$\mu = e^{\int \map f w \rd w}$ where $w$ depends on the nature of $\mu$.


First Order ODE/y dx + (x^2 y - x) dy = 0

The first order ODE: :$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$ has the general solution: :$\dfrac {y^2} 2 - \dfrac y x = C$


Set of Condensation Points of Countable Set is Empty

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :if $A$ is countable, :then $A^0 = \varnothing$.


Set of Condensation Points of Countable Set is Empty/Lemma

:if $A$ is countable, :then there exists no point $x$ of $S$ such that $x$ is a condensation point of $A$.


First Order ODE/(3 x^2 - y^2) dy - 2 x y dx = 0

The first order ODE: :$(1): \quad \paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$ has the general solution: :$\dfrac 1 y - \dfrac {x^2} {y^3} = C$


First Order ODE/y dx + x dy + 3 x^3 y^4 dy

The first order ODE: :$(1): \quad y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$ has the general solution: :$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$


Curved Mirror producing Parallel Rays is Paraboloid

Let $M$ be a curved mirror embedded in a real cartesian $3$- space. Let there be a source of light at the origin. Let $M$ reflect the light in a beam parallel to the $x$-axis. Then $M$ is the solid of revolution produced by rotating about the $x$-axis the parabola whose equation is: :$y^2 = 2 c x + c^2$


Quotient Rule for Differentials/Formulation 1

:$\map \d {\dfrac y x} = \dfrac {x \rd y - y \rd x} {x^2}$


Product Rule for Differentials

:$\map \d {x y} = x \rd y + y \rd x$


Differential of Sum of Squares

:$\map \d {x^2 + y^2} = 2 \paren {x \map \rd x + y \map \rd y}$


Differential of Arctangent of Quotient

:$\mathrm d \left({\arctan \dfrac x y}\right) = \dfrac{y \, \mathrm d x - x \, \mathrm d y} {x^2 + y^2}$


Differential of Logarithm of Quotient

:$\map \d {\ln \dfrac x y} = \dfrac {y \rd x - x \rd y} {x y}$


First Order ODE/y dx + (x^2 y - x) dy = 0/Proof 1

{{:First Order ODE/y dx + (x^2 y - x) dy = 0}} This can also be presented in the form: :$\dfrac {\d y} {\d x} + \dfrac y {x^2 y - x}$


First Order ODE/y dx - x dy = x y^3 dy

The first order ODE: :$(1): \quad y rd x - x \rd y = x y^3 \rd y$ has the general solution: :$\ln \dfrac x y = \dfrac {y^3} 3 + C$


First Order ODE/(x + y) dx = (x - y) dy/Proof 1

The first order ordinary differential equation: :$(1): \quad \paren {x + y} \rd x = \paren {x - y} \rd y$ is a homogeneous differential equation with solution: :$\arctan \dfrac y x = \ln \sqrt{x^2 + y^2} + C$


First Order ODE/x dy = (y + x^2 + 9 y^2) dx

The first order ODE: :$(1): \quad x \rd y = \paren {y + x^2 + 9 y^2} \rd x$ has the general solution: :$\map \arctan {\dfrac {3 y} x} = 3 x + C$


Linear First Order ODE/y' + (y over x) = 3 x

The linear first order ODE: :$\dfrac {\d y} {\d x} + \dfrac y x = 3 x$ has the general solution: :$x y = x^3 + C$ or: :$y = x^2 + \dfrac C x$


Linear First Order ODE/x y' - 3 y = x^4

The linear first order ODE: :$(1): \quad x \dfrac {\d y} {\d x} - 3y = x^4$ has the general solution: :$y = x^4 + \dfrac C {x^3}$


Linear First Order ODE/y' + y = 1 over (1 + exp 2 x)

The linear first order ODE: :$(1): \quad y' + y = \dfrac 1 {1 + e^{2 x} }$ has the general solution: :$y = e^{-x} \map \arctan {e^x} + C e^{-x}$


Linear First Order ODE/(1 + x^2) dy + 2 x y dx = cotangent x dx

The linear first order ODE: :$(1): \quad \paren {1 + x^2} \rd y + 2 x y \rd x = \cot x \rd x$ has the general solution: :$y = \dfrac {\map \ln {\sin x} } {1 + x^2} + \dfrac C {1 + x^2}$


Linear First Order ODE/(2 y - x^3) dx = x dy

The linear first order ODE: :$(1): \quad \paren {2 y - x^3} \rd x = x \rd y$ has the general solution: :$y = -x^3 + C x^2$


Solution to Bernoulli's Equation

'''Bernoulli's equation''': :$(1): \quad \dfrac {\d y} {\d x} + \map P x y = \map Q x y^n$ where: :$n \ne 0, n \ne 1$ has the general solution: :$\displaystyle \frac {\map \mu x} {y^{n - 1} } = \paren {1 - n} \int \map Q x \map \mu x \rd x + C$ where: :$\map \mu x = e^{\paren {1 - n} \int \map P x \rd x}$


Bernoulli's Equation/x y' + y = x^4 y^3

The first order ODE: :$(1): \quad x y' + y = x^4 y^3$ has the general solution: :$\dfrac 1 {y^2} = - x^4 + C x^2$


Topology Defined by Neighborhood System

Let $S$ be a set. Let $\family {\NN_x}_{x \mathop \in S}$ be an indexed family where $\NN_x$ is non-empty set of subsets of $S$. Assume that :$(\text N 1): \quad \forall x \in S, U \in \NN_x: x \in U$ :$(\text N 2): \quad \forall x \in S, U \in \NN_x, y \in U:\exists V \in \NN_y: V \subseteq U$ :$(\text N 3): \quad \forall x \in S, U_1, U_2 \in \NN_x: \exists U \in \NN_x: U \subseteq U_1 \cap U_2$ Then $T = \struct {S, \tau}$ is a topological space where: :$\tau = \displaystyle \set {\bigcup \GG: \GG \subseteq \bigcup_{x \mathop \in S} \NN_x}$ Moreover, $\family {\NN_x}_{x \mathop \in S}$ is a neighborhood system of $T$.


Orthogonal Trajectories/x + C exp -x

Consider the one-parameter family of curves: :$(1): \quad y = x + C e^{-x}$ Its family of orthogonal trajectories is given by the equation: :$x = y - 2 + C e^{-y}$ :600px


Solution of Second Order Differential Equation with Missing Dependent Variable

Let $\map f {x, y', y''} = 0$ be a second order ordinary differential equation in which the dependent variable $y$ is not explicitly present. Then $f$ can be reduced to a first order ordinary differential equation, whose solution can be determined.


Niemytzki Plane is Topology

Niemytzki plane is a topological space.


Solution of Second Order Differential Equation with Missing Independent Variable

Let $\map g {y, \dfrac {\d y} {\d x}, \dfrac {\d^2 y} {\d x^2} } = 0$ be a second order ordinary differential equation in which the independent variable $x$ is not explicitly present. Then $g$ can be reduced to a first order ordinary differential equation, whose solution can be determined.


Linear Second Order ODE/y'' + k^2 y = 0

The second order ODE: :$(1): \quad y'' + k^2 y = 0$ has the general solution: :$y = A \, \map \sin {k x + B}$ or can be expressed as: :$y = C_1 \sin k x + C_2 \cos k x$


Second Order ODE/y y'' + (y')^2 = 0

The second order ODE: :$(1): \quad y y'' + \paren {y'}^2 = 0$ has the general solution: :$y^2 = C_1 x + C_2$


Second Order ODE/x y'' = y' + (y')^3

The second order ODE: :$(1): \quad x y'' = y' + \paren {y'}^3$ has the general solution: :$x^2 + \paren {y - C_2}^2 = C_1^2$


Linear Second Order ODE/y'' - k^2 y = 0

The second order ODE: :$(1): \quad y'' - k^2 y = 0$ has the general solution: :$y = C_1 e^{k x} + C_2 e^{-k x}$