Parametric Equations for Evolute/Formulation 1

Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\begin{cases} X = x - \dfrac {y' \paren {1 + y'^2} } {y''} \\ Y = y + \dfrac {1 + y'^2} {y''} \end{cases}$ where: :$\left({x, y}\right)$ denotes the Cartesian coordinates of a general point on $C$ :$\left({X, Y}\right)$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $x$.


Parametric Equations for Evolute/Formulation 2

Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\begin{cases} X = x - \dfrac {y' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \\ Y = y + \dfrac {x' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \end{cases}$ where: :$\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$ :$\tuple {X, Y}$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$x'$ and $x''$ denote the derivative and second derivative respectively of $x$ {{WRT|Differentiation}} $t$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $t$.


Normal to Curve is Tangent to Evolute

Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $P$ be a point on $C$. Let $Q$ be the center of curvature of $C$ at $P$. The normal to $C$ at $P$ is tangent to the evolute $E$ of $C$ at $Q$.


Length of Arc of Evolute equals Difference in Radii of Curvature

Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. The length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$.


Curve is Involute of Evolute

Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $E$ be the evolute $C$. Then the involute of $E$ is $C$.


Evolute of Cycloid is Cycloid

The evolute of a cycloid is another cycloid.


Body behaves as Particle under Gravitation

A body of finite volume under the force of gravity behaves identically to a point mass under the same force.


Meet of Directed Subsets is Directed

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $D_1, D_2$ be directed subset of $S$. Then :$\left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$ is directed subset of $S$


Image of Directed Subset under Increasing Mapping is Directed

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be an increasing mapping. Let $D$ be a directed subset of $S$. Then :$f^\to \left({D}\right)$ is also a directed subset of $T$ where :$f^\to \left({D}\right)$ denotes the image of $D$ under $f$.


Mass of Sun from Gravitational Constant

Let the gravitational constant be known. Let the mean distance from the Earth to the sun be known. Then it is possible to calculate the mass of the sun.


Meet is Directed Suprema Preserving implies Meet of Suprema equals Supremum of Meet of Directed Subsets

Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $\struct {S \times S, \precsim}$ be the Cartesian product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Let $f: S \times S \to S$ be a mapping such that :$\forall s, t \in S: \map f {s, t} = s \wedge t$ and :$f$ preserves directed suprema. Let $D_1, D_2$ be directed subsets of $S$. Then :$\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {x \wedge y: x \in D_1, y \in D_2}$


Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving

Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice. Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that: :$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$ where  :$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$ Let :$\forall I_1, I_2 \in \map {\it Ids} {\mathscr S}: \paren {\sup I_1} \wedge \paren {\sup I_2} = \sup \set {i \wedge j: i \in I_1, j \in I_2}$ Then: $f$ preserves meet as a mapping from $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$ into $\mathscr S$


Field Norm of Quaternion is Positive Definite

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The field norm of $\mathbf x$: :$\map n {\mathbf x} := \cmod {\mathbf x \overline {\mathbf x} }$ is positive definite.


Field Norm of Quaternion is Multiplicative

Let $\mathbf x$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The field norm of $\mathbf x$: :$\map n {\mathbf x} := \size {\mathbf x \overline {\mathbf x} }$ is a multiplicative function.


Equivalence of Definitions of Oscillation at Point for Real Functions

Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. Let $x \in X$. {{TFAE|def = Oscillation/Real Space/Oscillation at Point|view = Oscillation at a Point}}


Supremum of Cartesian Product

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be an ordered set. Let $\left({S_1 \times S_2, \precsim}\right)$ be the Cartesian product of $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$. Let $X_1$ be a non-empty subset of $S_1$, $X_2$ be a non-empty subset of $S_2$ such that :$X_1$ and $X_2$ admit suprema. Then :$X_1 \times X_2$ admits a supremum and :$\sup \left({X_1 \times X_2}\right) = \left({\sup X_1, \sup X_2}\right)$


Nonassociative Division Algebra with Real Scalars has Dimension of Power of 2

Let $A$ be a nonassociative division algebra with real scalars. Then the dimension of $A$ is a power of $2$.


Bott-Milnor-Kervaire 1,2,4,8 Theorem

Let $A$ be a division algebra with real scalars. Then the dimension of $A$ is either: :$1$: the real numbers $\R$ :$2$: the complex numbers $\C$ :$4$: the quaternions $\Bbb H$ or: :$8$: the octonions $\Bbb O$.


Montel's Theorem

Let $U \subseteq \C$ be an open subset of the complex numbers. Let $\mathcal H \left({U}\right)$ be the space of holomorphic mappings on $U$. Then a family of mappings $\mathcal F \subseteq \mathcal H \left({U}\right)$ is normal {{iff}} $\mathcal F$ is locally bounded.


Vitali's Convergence Theorem

Let $U$ be an  open,  connected subset of $\C$. Let $S \subseteq U$ contain a limit point $\sigma$. Let $\sequence {f_n}_{n \mathop \in \N}$ be a  normal family of holomorphic mappings $f_n : U \to \C$. Let $\sequence {f_n}_{n \mathop \in \N}$  converge to some holomorphic mapping $f : U \to \C$ at $\sigma$. Then $f_n$ converges uniformly to $f$ on all compact subsets of $U$.


Identity Theorem

Let $U$ be an open connected subset of the complex plane $\C$. Let $f$ and $g$ be complex functions whose domain is $U$. Let $S = \left\{{z \in U: f \left({z}\right) = g \left({z}\right)}\right\}$. Let $f$ and $g$ be analytic on $U$. Let $S$ have a limit point in $U$. Then: :$\forall z \in U : f \left({z}\right) = g \left({z}\right)$


Exponential Sequence is Uniformly Convergent on Compact Sets

Let $\mathcal E = \left \langle{E_n}\right \rangle$ denote the sequence of functions $E_n: \C \to \C$ defined as: :$E_n \left({z}\right) = \left({1 + \dfrac z n}\right)^n$ Let $K$ be a compact subset of $\C$. Then $\mathcal E$ is uniformly convergent on $K$.


Uniformly Convergent Sequence Evaluated on Convergent Sequence

Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces. {{explain|Definition:Complete Metric Space is invoked, but it is called just "metric space". Analyse to determine whether it needs to be complete here or not.}} Let $K$ be a subspace of $X$. Let $\FF = \sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to Y$ uniformly convergent on $K$. Let $\sequence {a_n}$ be a Convergent Sequence in $K$ with limit $a \in K$. Then $\sequence {\map {f_n} {a_n} }$ is convergent such that: :$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} {a_n} = \map f a$


Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Ideals

Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Then :$\mathscr S$ is meet-continuous {{iff}} :for every ideals $I, J$ in $\mathscr S$: $\paren {\sup I} \wedge \paren {\sup J} = \sup \set {i \wedge j: i \in I, j \in J}$


Ellipse is Mechanical Curve

The ellipse is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given ellipse.


Parabola is Mechanical Curve

The parabola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given parabola.


Hyperbola is Mechanical Curve

The hyperbola is a mechanical curve. That is, it cannot be constructed using a compass and straightedge. However, as many points as is desired can be so constructed that lie on any given hyperbola.


Derivative of Sequence of Holomorphic Functions

Let $U$ be an  open,  connected subset of $\C$. Let $\left \langle{f_n}\right \rangle$ be a sequence of  holomorphic mappings $f_n: U \to \C$. Let $\left \langle{f_n}\right \rangle$  converge pointwise to some function $f: U \to \C$. Let $\left \langle{f_n}\right \rangle$  converge uniformly on  compact subsets of $U$. Then $f$ is holomorphic on $U$. Further, the sequence of  derivatives $\left \langle{f_n'}\right \rangle$ converges to $f'$ on $U$.  This convergence is uniform on compact subsets of $U$.


Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded

Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left \langle{f_i}\right \rangle_{i \in I}$ be a uniformly convergent sequence of mappings $f_i: X \to Y$. $\forall i \in I$, let $f_i$ be bounded. Then $\left \langle{f_i}\right \rangle$ is uniformly bounded.


Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces. Let $\sequence {f_n}$ and $\sequence {g_n}$ be sequences of mappings from $X$ to $Y$. Let $\sequence {f_n}$ and $\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.  $\forall n \in \N$, let $f_n$ and $g_n$ be bounded. Then the sequence $\sequence {f_n g_n}$ is uniformly convergent on $S$. {{explain|for the product to be defined, we probably want $Y$ to have some more structure}}


Area between Radii and Curve in Polar Coordinates

Let $C$ be a curve expressed in polar coordinates $\polar {r, \theta}$ as: :$r = \map g \theta$ where $g$ is a real function. Let $\theta = \theta_a$ and $\theta = \theta_b$ be the two rays from the pole at angles $\theta_a$ and $\theta_b$ to the polar axis respectively. Then the area $\mathcal A$ between $\theta_a$, $\theta_b$ and $C$ is given by: :$\displaystyle \mathcal A = \int \limits_{\theta \mathop = \theta_a}^{\theta \mathop = \theta_b} \frac {\paren {\map g \theta}^2 \rd \theta} 2$ as long as $\paren {\map g \theta}^2$ is integrable.


Derivative of Exponential Function/Complex

The complex exponential function is its own derivative. That is: :$\map {D_z} {\exp z} = \exp z$


Exponential Sequence is Eventually Increasing

Let $\sequence {E_n}$ be the sequence of real functions $E_n: \R \to \R$ defined as: :$\map {E_n} x = \paren {1 + \dfrac x n}^n$ Then, for sufficiently large $n \in \N$, $\sequence {\map {E_n} x}$ is increasing with respect to $n$. That is: :$\forall x \in \R: \forall n \in \N: n \ge \ceiling {\size x} \implies \map {E_n} x \le \map {E_{n + 1} } x$ where $\ceiling x$ denotes the ceiling of $x$.


Euler's Number: Limit of Sequence implies Base of Logarithm

Let $e$ be  Euler's number defined by: :$\displaystyle e := \lim_{n \mathop \to \infty} \left({1 + \frac 1 n}\right) ^n$ Then $e$ is the unique solution to the equation $\ln \left({x}\right) = 1$. That is: :$\ln \left({x}\right) = 1 \iff x = e$


Diophantine Equation y cubed equals x squared plus 2

The indeterminate Diophantine equation: :$y^3 = x^2 + 2$ has only one integer solution: :$x = 5, y = 3$


Continuous Extension from Dense Subset

Let $X$ be a metric space. Let $D$ be a dense subset of $X$. Let $f: D \to \R$ be a uniformly continuous mapping. Then there exists a unique continuous extension of $f$ to $X$.


Sum of two Fourth Powers cannot be Fourth Power

$\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions.


Moore-Osgood Theorem

Let $X$ and $Y$ be metric spaces. Let $S$ be a subspace of $X$. Let $c$ be a limit point of $S$. Let $\left \langle{f_n}\right \rangle$ be a sequence of mappings $f_n : X \to Y$. Suppose that: :$(1): \quad \left \langle{f_n}\right \rangle$ is uniformly convergent on $S$ :$(2): \quad \displaystyle \forall n \in \N : \lim_{x \to c} f_n \left({ x }\right)$ exists Then: :$\displaystyle \lim_{x \to c} \lim_{n \to \infty} f_n \left({ x }\right) = \lim_{n \to \infty} \lim_{x \to c} f_n \left({ x }\right)$


Cycloid is Best Shape for Arch

The optimal shape for an arch is a cycloid.


Center of Gravity of Cycloid

Let $C$ be a cycloid. The center of gravity of an arc of $C$ is at {{stub|research needed}}


Ore Graph is Connected

Let $G = \left({V, E}\right)$ be an Ore graph. Then $G$ is connected.


Power Function to Rational Power permits Unique Continuous Extension

Let $a \in \R_{> 0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\R$.


Meet-Continuous iff Meet of Suprema equals Supremum of Meet of Directed Subsets

Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Then: :$\mathscr S$ is meet-continuous {{iff}}: :for every directed subsets $D_1, D_2$ of $S$: $\paren {\sup D_1} \wedge \paren {\sup D_2} = \sup \set {d_1 \wedge d_2: d_1 \in D_1, d_2 \in D_2}$


Brouwerian Lattice iff Meet-Continuous and Distributive

Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Then :$\mathscr S$ is a Brouwerian lattice {{iff}} :$\mathscr S$ is meet-continuous and distributive.


Power is Well-Defined/Integer

Let $x$ be a non-zero real number. Let $k$ be an integer. Then $x^k$ is well-defined.


Plane Figure with Maximum Area for given Perimeter is Circle

Let $F$ be a plane figure. Let $P$ be the length of the perimeter of $F$. Let the area of $F$ be the largest of all the plane figures whose perimeters are of length $P$. Then $F$ is a circle.


Real Number between Zero and One is Greater than Power/Natural Number

Let $x \in \R$. Let $0 < x < 1$. Let $n$ be a natural number. Then: : $0 < x^n \le x$


Power Function is Strictly Increasing over Positive Reals/Natural Exponent

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $f: \R_{>0} \to \R$ be the real function defined as: :$f \left({x}\right) = x^n$ where $x^n$ denotes $x$ to the power of $n$. Then $f$ is strictly increasing.


Root is Strictly Increasing

Let $x \in \R_{> 0}$. Let $n \in \N$. Let $f: \R_{> 0} \to \R$ be the real function defined as: :$\map f x = \sqrt [n] x$ where $\sqrt [n] x$ denotes the $n$th root of $x$. Then $f$ is strictly increasing.


Shift Mapping is Lower Adjoint iff Meet is Distributive with Supremum

Let $\left({S, \preceq}\right)$ be a complete lattice. Then :$\forall x \in S, f: S \to S:\left({\forall y \in S: f\left({y}\right) = x \wedge y}\right) \implies f$ is lower adjoint of Galois connection {{iff}} :$\forall x \in S, X \subseteq S: x \wedge \sup X = \sup \left\{ {x \wedge y: y \in X}\right\}$


Greatest Common Divisor is Associative

Let $a, b, c \in \Z$. Then: :$\gcd \left\{ {a, \gcd \left\{{b, c}\right\} }\right\} = \gcd \left\{ {\gcd \left\{{a, b}\right\}, c}\right\}$ where $\gcd$ denotes the greatest common divisor.


Lowest Common Multiple is Associative

Let $a, b, c \in \Z$. Then: :$\lcm \left\{{a, \lcm \left\{{ b , c }\right\} }\right\} = \lcm \left\{{\lcm \left\{{a, b}\right\}, c}\right\}$ where $\lcm$ denotes the lowest common multiple.


LCM of Coprime Integers

Let $a, b \in \Z_{>0}$ be coprime integers. Then: :$\lcm \set {a, b} = a b$ where $\lcm$ denotes the lowest common multiple.


GCD with One Fixed Argument is Multiplicative Function

Let $a, b, c \in \Z: b \perp c$ where $b \perp c$ denotes that $b$ is coprime to $c$. Then: :$\gcd \set {a, b} \gcd \set {a, c} = \gcd \set {a, b c}$ That is, GCD is multiplicative.


Power Function on Base between Zero and One is Strictly Decreasing/Positive Integer

Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as: :$\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly decreasing.


Power Function on Base between Zero and One is Strictly Decreasing/Rational Number

Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly decreasing.


Power Function on Base Greater than One is Strictly Increasing/Positive Integer

Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as: :$\map f n = a^n$ where $a^n$ denotes $a$ to the power of $n$. Then $f$ is strictly increasing.


Root of Reciprocal is Reciprocal of Root

Let $x \in \R_{\ge 0}$. Let $n \in \N$. Let $\sqrt [n] x$ denote the $n$th root of $x$. Then: :$\sqrt [n] {\dfrac 1 x} = \dfrac 1 {\sqrt [n] x}$


Power Function on Base Greater than One is Strictly Increasing/Rational Number

Let $a \in \R$ be a real number such that $a > 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then $f$ is strictly increasing.


Existence of Square Roots of Positive Real Number

Let $r \in \R_{\ge 0}$ be a positive real number. Then: :$\exists y_1 \in \R_{\ge 0}: {y_1}^2 = r$ :$\exists y_2 \in \R_{\le 0}: {y_2}^2 = r$


Inequality iff Difference is Positive

Let $x, y \in \R$. Then the following are equivalent: :$(1): \quad x < y$ :$(2): \quad y - x > 0$


Power Function on Base between Zero and One is Strictly Decreasing/Integer

Let $a \in \R$ be a real number such that $0 < a < 1$. Let $f: \Z \to \R$ be the real-valued function defined as: :$\map f k = a^k$ where $a^k$ denotes $a$ to the power of $k$. Then $f$ is strictly decreasing.


Meet-Continuous iff Meet Preserves Directed Suprema

Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $\left({S \times S, \precsim}\right)$ be the Cartesian product of $\mathscr S$ and $\mathscr S$. Let $f: S \times S \to S$ be a mapping such that :$\forall x, y \in S: f\left({x, y}\right) = x \wedge y$ Then :$\mathscr S$ is meet-continuous {{iff}} :$f$ preserves directed suprema


Real Star-Algebra is Commutative

Let $A = \left( A_F , \oplus \right)$ be a real *-algebra whose conjugation is denoted as $*$. Then: :$\forall a,b \in A, a \oplus b = b \oplus a$ That is, real *-algebra is commutative.


Meet-Continuous implies Shift Mapping Preserves Directed Suprema

Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous lattice. Let $x \in S$. Let $f: S \to S$ be a mapping such that :$\forall y \in S: f\left({y}\right) = x \wedge y$ Then :$f$ preserves directed suprema.


Power of Strictly Positive Real Number is Strictly Positive/Positive Integer

Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$.


Power of Positive Real Number is Positive/Integer

Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z$ be an integer. Then: :$x^n > 0$ where $x^n$ denotes the $n$th power of $x$.


Power of Positive Real Number is Positive/Rational Number

Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $q \in \Q$ be a rational number. Then: :$x^q > 0$ where $x^q$ denotes the $x$ to the power of $q$.


Power Function is Monotone/Rational Number

Let $a \in \R_{>0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then $f$ is monotone. Further, $f$ is strictly monotone unless $a = 1$.


Power Function tends to One as Power tends to Zero/Rational Number

Let $a \in \R_{> 0}$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f q = a^q$ where $a^q$ denotes $a$ to the power of $q$. Then: :$\displaystyle \lim_{x \mathop \to 0} \map f x = 1$


Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma

Let $a \in \R$ be a real number such that $a > 1$. Let $r \in \Q_{> 0}$ be a strictly positive rational number such that $r < 1$. Then: :$1 < a^r < 1 + a r$


Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number

Let $a \in \R_{> 0}$ be a strictly positive real number such that $a > 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\displaystyle \lim_{r \mathop \to 0} \map f r = 1$


Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number

Let $a \in \R_{> 0}$ be a strictly positive real number such that $0 < a < 1$. Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f r = a^r$ where $a^r$ denotes $a$ to the power of $r$. Then: :$\displaystyle \lim_{r \mathop \to 0} \map f r = 1$


Euler's Number to Rational Power permits Unique Continuous Extension

Let $e$ be Euler's number. Let $f: \Q \to \R$ be the real-valued function defined as: :$f \left({q}\right) = e^q$ where $e^q$ denotes  $e$ to the power of $q$. Then there exists a unique continuous extension of $f$ to $\R$.


Open Unit Interval is Proper Subset of Closed Unit Interval

The open unit interval: :$I_o = \left({0 \,.\,.\, 1}\right)$ is a proper subset of the closed unit interval: :$I_c = \left[{0 \,.\,.\, 1}\right]$


Vector Cross Product satisfies Jacobi Identity

Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in $3$ dimensional Euclidean space. Let $\times$ denotes the cross product. Then: :$\mathbf a \times \paren {\mathbf b \times \mathbf c} + \mathbf b \times \paren {\mathbf c \times \mathbf a} + \mathbf c \times \paren {\mathbf a \times \mathbf b} = \mathbf 0$ That is, the cross product operation satisfies the Jacobi identity.


Meet-Continuous and Distributive implies Shift Mapping Preserves Finite Suprema

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous distributive complete lattice. Let $x \in S$. Let $f: S \to S$ be a mapping such that :$\forall y \in S: f \left({y}\right) = x \wedge y$ Then :$f$ preserves finite suprema


Set Union is not Cancellable

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


Intersection of Real Intervals is Real Interval

Let $I_1$ and $I_2$ be real intervals. Then $I_1 \cap I_2$ is also a real interval.


Union of Real Intervals is not necessarily Real Interval

Let $I_1$ and $I_2$ be real intervals. Then $I_1 \cup I_2$ is not necessarily a real interval.


Pasting Lemma/Corollary 1

Let $A$ and $B$ be closed in $X$. Let $f : A \to Y$ and $g : B \to Y$ be continuous mappings that agree on $A \cap B$. Then the mapping $f \cup g : A \cup B \to Y$ is continuous.


Pasting Lemma/Corollary 2

Let $\mathcal A = \left\{ {A_i: i \in I} \right\}$ be a set of sets that are open in $X$. Let $f: \bigcup \mathcal A \to Y$ be a mapping such that: :$\forall i \in I : f \restriction A_i$ is continuous Then $f$ is continuous on $\bigcup \mathcal A$.


Pasting Lemma

Let $X$ and $Y$ be topological spaces. Let $A$ and $B$ be open in $X$. Let $f: A \to Y$ and $g: B \to Y$ be continuous mappings that agree on $A \cap B$. Let $f \cup g$ be the union of the mappings $f$ and $g$: :$\forall x \in A \cup B: \map {f \cup g} x = \begin {cases} \map f x & : x \in A \\ \map g x & : x \in B \end {cases}$ Then the mapping $f \cup g : A \cup B \to Y$ is continuous.


Domain of Real Square Root Function

The domain of the real square root function is the set of positive real numbers $\R_{\ge 0}$: :$\left\{ {x \in \R: x \ge 0}\right\}$


Image of Real Square Root Function

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


Exponent Combination Laws/Product of Powers/Proof 1

Let $a \in \R_{> 0}$ be a positive real number. Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^x a^y = a^{x + y}$


Inequality of Product of Unequal Numbers

Let $a, b, c, d \in \R$. Then: :$0 < a < b \land 0 < c < d \implies 0 < a c < b d$


Negative of Absolute Value/Corollary 3

Let $y \in \R_{\ge 0}$. Let $z \in \R$. Then: :$\left\vert{x - z}\right\vert < y \iff z - y < x < z + y$


Exponent Combination Laws/Product of Powers/Proof 2/Lemma

Let $x_1, x_2, y_1, y_2 \in \R_{>0}$ be strictly positive real numbers. Let $\epsilon \in \openint 0 {\min \set {y_1, y_2, 1} }$. Then: :$\size {x_1 - y_1} < \epsilon \land \size {x_2 - y_2} < \epsilon \implies \size {x_1 x_2 - y_1 y_2} < \epsilon \paren {y_1 + y_2 + 1}$


Union of Unordered Tuples

Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary. Then :$\set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m} = \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$


Gauss-Lucas Theorem

Let $P$ be a (nonconstant) polynomial with complex coefficients. Then all zeros of its derivative $P'$  belong to the convex hull of the set of zeros of $P$.


Way Below implies Preceding

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x \ll y$ where $\ll$ denotes element is way below second element. Then :$x \preceq y$


Preceding and Way Below implies Way Below

Let $\left({S, \preceq}\right)$ be an ordered set. Let $u, x, y, z \in S$ such that :$u \preceq x \ll y \preceq z$ where $\ll$ denotes the way below relation. Then :$u \ll z$


Join is Way Below if Operands are Way Below

Let $\left({S, vee, \preceq}\right)$ be a join semilattice. Let $x, y, z \in S$ such that :$x \ll z$ and $y \ll z$ where $\ll$ denotes the way below relation. Then :$x \vee y \ll z$


Way Below Relation is Transitive

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y, z \in S$ such that :$x \ll y \ll z$ Then :$x \ll z$


Way Below Relation is Antisymmetric

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x \ll y$ and $y \ll x$ Then :$x = y$


Continued Fraction Expansion of Golden Mean

The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$: :$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$


Fibonacci Number less than Golden Section to Power less One

For all $n \in \N_{> 0}$: :$F_n \le \phi^{n - 1}$ where: :$F_n$ is the $n$th Fibonacci number :$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$


Square of Golden Mean equals One plus Golden Mean

:$\phi^2 = \phi + 1$ where $\phi$ denotes the golden mean.


Fibonacci Number greater than Golden Section to Power less Two

For all $n \in \N_{\ge 2}$: :$F_n \ge \phi^{n - 2}$ where: :$F_n$ is the $n$th Fibonacci number :$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$


Triangular Number as Alternating Sum and Difference of Squares

{{begin-eqn}} {{eqn | ll= \forall n \in \N:        | l = \frac {n \paren {n + 1} } 2       | r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2       | c =  }} {{eqn | r = n^2 - \paren {n - 1}^2 + \paren {n - 2}^2 - \cdots + \paren {-1}^{n - 1}        | c =  }} {{end-eqn}} Thus the $n$th triangular number can be expressed as the alternating sum and difference of squares: So: {{begin-eqn}} {{eqn | l = 1       | r = 1^2       | c =  }} {{eqn | l = 3       | r = 2^2 - 1^2       | c =  }} {{eqn | l = 6       | r = 3^2 - 2^2 + 1^2       | c =  }} {{eqn | l = 10       | r = 4^2 - 3^2 + 2^2 - 1^2       | c =  }} {{end-eqn}} and so on.


Bernoulli's Inequality/Corollary

Let $x \in \R$ be a real number such that $0 < x < 1$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\left({1 - x}\right)^n \ge 1 - n x$


2 to the n is Greater than n Cubed when n is 10 and above

:$\forall n \in \Z, n \ge 10: 2^n > n^3$


Set of Integers can be Well-Ordered

The set of integers $\Z$ can be well-ordered with an appropriately chosen ordering.


Existence of Digital Root

Let $n \in \N$ be a natural number. Let $b \in \N$ such that $b \ge 2$ also be a natural number. Let $n$ be expressed in base $b$. Then the digital root base $b$ exists for $n$.


Golden Mean is Irrational

The golden mean $\phi$ is irrational.


Way Below iff Preceding Finite Supremum

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$. Then $x \ll y$ {{iff}} :$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in {\it Fin}\left({X}\right): x \preceq \sup A$ where :$\ll$ denotes the way below relation, :${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Exponent Combination Laws/Product of Powers/Proof 2

Let $a \in \R_{> 0}$ be a positive real number. Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: : $a^x a^y = a^{x + y}$ {{proofread}}


Exponent Combination Laws/Power of Power/Proof 1

Let $a \in \R_{>0}$ be a (strictly) positive real number. {{:Exponent Combination Laws/Power of Power}}


Exponent Combination Laws/Power of Power/Proof 2


Rational Sequence Increasing to Real Number

Let $x \in \R$ be a real number. Then there exists some increasing rational sequence that converges to $x$.


Rational Sequence Decreasing to Real Number

Let $x \in \R$ be a real number. Then there exists some decreasing rational sequence that converges to $x$.


Power Function on Base Greater than One is Strictly Increasing/Real Number

Let $a \in \R$ be a real number such that $a > 1$. Let $f: \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is strictly increasing.


Power Function on Strictly Positive Base is Continuous/Real Power

Let $f : \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous.


Power Function on Strictly Positive Base is Continuous/Rational Power

Let $f: \Q \to \R$ be the real-valued function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is continuous.


Defining Sequence of Natural Logarithm is Convergent

Let $x \in \R$ be a real number such that $x > 0$. Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$f_n \left({ x }\right) = n \left({ \sqrt[n]{ x } - 1 }\right)$ Then $\left\langle{ f_n }\right\rangle$ is pointwise convergent.


Exponent Combination Laws/Positive Integers

Let $r \in \R_{>0}$ be (strictly) positive real numbers.


Sum of Indices of Real Number/Rational Numbers

Let $x, y \in \Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $n$. Then: : $r^{x + y} = r^x \times r^y$


Exponent Combination Laws/Rational Numbers

Let $r \in \R_{> 0}$ be a (strictly) positive real number.


Product of Indices of Real Number/Rational Numbers

Let $x, y \in \Q$ be rational numbers. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\paren {r^x}^y = r^{x y}$


Sum of Indices of Real Number


Product of Indices of Real Number


Sum of Indices of Real Number/Integers

Let $n, m \in \Z$ be integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{n + m} = r^n \times r^m$


Product of Indices of Real Number/Integers

Let $n, m \in \Z$ be positive integers. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$\paren {r^n}^m = r^{n m}$


Uniformly Convergent iff Difference Under Supremum Metric Vanishes

Let $X$ and $Y$ be metric spaces Let $\sequence {f_n}$ be a sequence of mappings defined on $X$. Let $f: X \to Y$ be a mapping. Let $d_S: S \times S \to Y$ denote the supremum metric on $S \subseteq X$. Then $\sequence {f_n}$ converges uniformly to $f$ on $S$ {{iff}} $\map {d_S} {f_n, f} \to 0$ as $n \to \infty$.


Uniform Convergence is Hereditary

Let $M = \struct {A, d}$ be a metric space. Let $\sequence {f_n}$ be a sequence of mappings defined on $A$. Let $\sequence {f_n}$ be uniformly convergent on $S \subseteq A$. Then $\sequence {f_n}$ is uniformly convergent on every metric subspace of $S$. That is, uniform convergence is a hereditary property of a metric space.


Real Number to Negative Power


Real Number to Negative Power/Integer

Let $n \in \Z$ be an integer. Let $r^n$ be defined as $r$ to the power of $n$. Then: :$r^{-n} = \dfrac 1 {r^n}$


Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal

Let $\mathscr S = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $x, y \in S$. Then $x \ll y$ {{iff}} :$\forall I \in {\it Ids}\left({\mathscr S}\right): y \preceq \sup I \implies x \in I$ where :$\ll$ denotes the way below relation, :${\it Ids}\left({\mathscr S}\right)$ denotes the set of all ideals in $\mathscr S$.


Lower Closure of Directed Subset is Ideal

Let $\mathscr S = \left({S, \preceq}\right)$ be an ordered set. Let $D$ be a directed subset of $S$. Then :$D^\preceq$ is an ideal in $\mathscr S$ where $D^\preceq$ denotes the lower closure of $D$.


Sum of Geometric Sequence/Corollary 2

:$\displaystyle \sum_{j \mathop = 0}^{n - 1} j x^j = \frac {\paren {n - 1} x^{n + 1} - n x^n + x} {\paren {x - 1}^2}$


Defining Sequence of Natural Logarithm is Strictly Decreasing

Let $x \in \R$ be a real number such that $x > 0$. Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Then $\forall x \in \R_{>0}: \sequence {\map {f_n} x}$ is strictly decreasing.


Logarithm Base 10 of 2 is Irrational

The common logarithm of $2$: :$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$ is irrational.


Nth Root of 1 plus x not greater than 1 plus x over n

Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$\sqrt [n] {1 + x} \le 1 + \dfrac x n$


Reciprocal of Logarithm

Let $x, y \in \R_{> 0}$ be (strictly) positive real numbers. Then: :$\dfrac 1 {\log_x y} = \log_y x$


Discontinuity of Monotonic Function is Jump Discontinuity

Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a monotone real function. Then $f$ is discontinuous at a point $c \in X$ {{iff}} $c$ is a jump discontinuity of $f$.


Surjective Monotone Function is Continuous

Let $X$ be an open subset of $\R$. Let $Y$ be a real interval. Let $f: X \to Y$ be a surjective monotone real function. Then $f$ is continuous on $X$.


Limit of Bounded Convergent Sequence is Bounded

Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ be convergent sequences in $\R$. Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ converge to $x, a, b \in \R$, respectively. Suppose that: :$\exists N \in \N : n \geq N \implies a_n \leq x_n \leq b_n$ Then: :$ a \le x \le b$


Logarithm of Power/Natural Logarithm/Natural Power

Let $x \in \R$ be a strictly positive real number. Let $n \in \R$ be any natural number. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({x^n}\right) = n \ln x$


Logarithm of Power/Natural Logarithm/Integer Power

Let $x \in \R$ be a strictly positive real number. Let $n \in \R$ be any integer. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({ x^n }\right) = n \ln x$


Logarithm of Power/Natural Logarithm/Rational Power

Let $x \in \R$ be a strictly positive real number. Let $r \in \R$ be any rational number. Let $\ln x$ be the natural logarithm of $x$. Then: :$\ln \left({ x^r }\right) = r \ln x$


Dini's Theorem

Let $K \subseteq \R$ be compact. Let $\sequence {f_n}$ be a sequence of continuous real functions defined on $K$. Let $\sequence {f_n}$ converge pointwise to a continuous function $f$. Suppose that: :$\forall x \in K : \sequence {\map {f_n} x}$ is monotone. Then the convergence of $\sequence {f_n}$ to $f$ is uniform.


Way Below in Meet-Continuous Lattice

Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous bounded below lattice. Let $x, y \in S$. Then $x \ll y$ {{iff}} :$\forall I \in {\it Ids}\left({\mathscr S}\right): y = \sup I \implies x \in I$ where :$\ll$ denotes the way below relation, :${\it Ids}\left({\mathscr S}\right)$ denotes the set of all ideals in $\mathscr S$.


Approximation to Binary Logarithm from Natural and Common Logarithm

The binary logarithm $\lg x$ can be approximated, to within $1 \%$, by the expression: :$\lg x \approx \ln x + \log_{10} x$ That is, by the sum of the natural logarithm and common logarithm.


Monotone Real Function with Everywhere Dense Image is Continuous

Let $I$ and $J$ be real intervals. Let $f: I \to J$ be a monotone real function. Let $f \sqbrk I$ be everywhere dense in $J$, where $f \sqbrk I$ denotes the image of $I$ under $f$. Then $f$ is continuous on $I$.


Relative Difference between Infinite Set and Finite Set is Infinite

Let $S$ be an infinite set. Let $T$ be a finite set. Then $S \setminus T$ is an infinite set.


Set Difference over Subset

Let $A$, $B$, and $S$ be sets. Let $A \subseteq B$. Then: :$A \setminus S \subseteq B \setminus S$


Subset of Empty Set iff Empty

Let $S$ be a set. Let $\O$ denote the empty set. Then $S \subseteq \O$ {{iff}} $S = \O$.


Finite Subset of Metric Space is Closed

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be finite. Then $S$ is closed in $M$.


Monotone Real Function with Everywhere Dense Image is Continuous/Lemma

:$\displaystyle \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c+ } \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$


Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets

Let $x \in \R$ be a real number such that $x > 0$. Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Let $K \subseteq \R_{>0}$ be compact. Then $\sequence {f_n}$ is uniformly convergent on $K$.


Multiplication of Positive Number by Real Number Greater than One

Let $x$ and $y$ be real numbers. Let $x > 1$. Let $y > 0$. Then $\dfrac y x < y$.


Set of Finite Suprema is Directed

Let $\left({S, \vee, \preceq}\right)$ be a join semilattice. Let $X$ be a non-empty subset of $S$. Then :$\left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$ is directed. where ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Change of Index Variable of Summation

:$\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({j}\right)} a_j$


Singleton is Finite

Let $x$ be arbitrary. Then $\set x$ is a finite set.


T4 Property Preserved in Closed Subspace/Corollary

If $T$ is a normal space then $T_K$ is also a normal space. That is, the property of being a normal space is weakly hereditary.


Derivative of Uniformly Convergent Sequence of Differentiable Functions

Let $J$ be a bounded interval. Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions $f_n: J \to \R$. Let each of $\left\langle{f_n}\right\rangle$ be differentiable on $J$. Let $\left\langle{ f_n \left({ x_0 }\right) }\right\rangle$ be convergent for some $x_0 \in J$. Let the sequence of derivatives $\left\langle{f_n'}\right\rangle$ converge uniformly on $J$ to a function $g : J \to \R$. Then $\left\langle{f_n}\right\rangle$ converge uniformly on $J$ to a differentiable function $f: J \to \R$ and $D_x f = g$.


Derivative of Natural Logarithm Function/Proof 4/Lemma

Let $\sequence {f_n}_n$ be the sequence of real functions $f_n: \R_{>0} \to \R$ defined as: :$\map {f_n} x = n \paren {\sqrt [n] x - 1}$ Let $k \in \N$. Let $J = \closedint {\dfrac 1 k} k$. Then the sequence of derivatives $\sequence { {f_n}'}_n$ converges uniformly to some real function $g: J \to \R$.


Exponential Function is Well-Defined/Real

Let $x \in \R$ be a real number. Let $\exp x$ be the exponential of $x$. Then $\exp x$ is well-defined.


Tail of Convergent Sequence

Let $\left\langle{a_n}\right\rangle$ be a real sequence. Let $N \in \N$ be a natural number. Let $a \in R$ be a real number. Then: :$a_n \to a$ {{iff}}: :$a_{n + N} \to a$


Union of Inverses of Mappings is Inverse of Union of Mappings

Let $I$ be an indexing set. Let $\family {f_i: i \in I}$ be an indexed family of mappings. For each $i \in I$, let $f^{-1}$ denote the inverse of $f$. Then the inverse of the union of $\family {f_i: i \in I}$ is the union of the inverses of $f_i, i \in I$. That is: :$\ds \paren {\bigcup \family {f_i: i \in I} }^{-1} = \bigcup \family {f_i^{-1}: i \in I}$


Union of Functions Theorem/Corollary

For each $i \in \N$, let $g_i : X_i \to Y$ be invertible. Then $\displaystyle \bigcup \left\{{g_i: i \in \N}\right\}$ is invertible and: :$\displaystyle \left({\bigcup \left\{{g_i: i \in \N}\right\} }\right)^{-1} = \bigcup \left\{{g_i^{-1}: i \in \N}\right\}$


Permutation of Indices of Summation

:$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$


Translation of Index Variable of Summation

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{R \left({j}\right)} a_j$ denote a summation over $R$. Then: :$\displaystyle \sum_{R \left({j}\right)} a_j = \sum_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \sum_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$.


Axiom of Approximation in Up-Complete Semilattice

Let $\mathscr S = \left({S, \wedge, \preceq}\right)$ be an up-complete meet semilattice. Let :$\forall x \in S: x^\ll$ is directed Then :$\mathscr S$ satisfies axiom of approximation {{iff}} :$\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$


Operand is Upper Bound of Way Below Closure

Let $\struct {S, \preceq}$ be an ordered set. Let $x \in S$. Then :$x$ is upper bound for $x^\ll$ where $x^\ll$ denotes the way below closure of $x$.


Exchange of Order of Summation

:$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Sum of Summations equals Summation of Sum/Infinite Sequence

Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers $\Z$. Let $\displaystyle \sum_{\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Let $\displaystyle \sum_{\map R i} b_i$ and $\displaystyle \sum_{\map R i} c_i$ be convergent. Then: :$\displaystyle \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$


Exchange of Order of Summation with Dependency on Both Indices

:$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$ where: :$S' \left({j}\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold :$R' \left({i, j}\right)$ denotes the propositional function: ::both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.


Summation of i from 1 to n of Summation of j from 1 to i

:$\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^i a_{i j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = j}^n a_{i j}$


Exchange of Order of Summation/Infinite Series

Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} \left\vert{a_{i j} }\right\vert$ exist. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Exchange of Order of Summation with Dependency on Both Indices/Infinite Series

Let the fiber of truth of both $R$ and $S$ be infinite. Let: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} \left\vert{a_{i j} }\right\vert$ exist. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({i, j}\right)} a_{i j} = \sum_{S' \left({j}\right)} \sum_{R' \left({i, j}\right)} a_{i j}$ where: :$S' \left({j}\right)$ denotes the propositional function: ::there exists an $i$ such that both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold :$R' \left({i, j}\right)$ denotes the propositional function: ::both $R \left({i}\right)$ and $S \left({i, j}\right)$ hold.


Exchange of Order of Summation/Finite and Infinite Series

Let the fiber of truth of $R$ be infinite. Let the fiber of truth of $S$ be finite. For all $j$ in the fiber of truth of $S$, let $\displaystyle \sum_{R \left({i}\right)} a_{i j}$ be convergent. Then: :$\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$


Permutation of Indices of Summation/Infinite Series

Let the fiber of truth of $R$ be infinite. Let $\displaystyle \sum_{\map R i} a_i$ be absolutely convergent. Then: :$\displaystyle \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$


Sum of Summations equals Summation of Sum

Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{\map R i} x_i$ denote a summation over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$


Sum of Summations over Overlapping Domains

:$\displaystyle \sum_{R \left({j}\right)} a_j + \sum_{S \left({j}\right)} a_j = \sum_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j + \sum_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.


Rules for Manipulating Summations

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers. Let $S: \Z \times \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional functions on the Cartesian product of the set of integers with itself. Let $\displaystyle \sum_{R \left({i}\right)} x_i$ denote a summation over $R$. Let $\pi$ be a permutation on the fiber of truth of $R$.


Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Let $D \subseteq \R^2$ be a region containing $\tuple {a, b}$. Let $f: D \to \R$ be real-valued mapping such that $f$ and $\dfrac {\partial f} {\partial x}$ are continuous on $D$. Consider the initial value problem: :$\dfrac {\d x} {\d t} = \map f {t, x}$ :$\map x a = b$ Suppose the initial value problem above has a solution $\phi$ for all $x$ in some interval $J$ containing $a$. Then the solution $\phi$ is  unique on $J$.


Exhausting Sequence of Sets on the Strictly Positive Real Numbers

For each $k \in \N$, let $S_k = \openint {\dfrac 1 k} k$. Then $\sequence {S_k}_k$ is an exhausting sequence of sets on $\R_{>0}$.


Way Below in Ordered Set of Topology

Let $\left({S, \tau}\right)$ be a topological space. Let $\left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$ Let $x, y \in \tau$. Then :$x \ll y$ in $\left({\tau, \preceq}\right)$ {{iff}} :for every set $F$ of open subsets of $S$: if $y \subseteq \bigcup F$, then ::there exists a finite subset $G$ of $F$: $x \subseteq \bigcup G$ where $\ll$ denotes the way below relation.


Continuous Inverse Theorem

Let $f$ be a real function defined on an interval $I$. Let $f$ be strictly monotone and continuous on $I$. Let $g$ be the inverse mapping to $f$. Let $J := f \left[{I}\right]$ be the image of $I$ under $f$. Then $g$ is strictly monotone and continuous on $J$.


Logarithm is Strictly Concave

:$\ln x: x > 0$ strictly concave.


Exponential is Strictly Convex

:The function $f \left({x}\right) = \exp x$ is strictly convex.


Way Below in Complete Lattice

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x, y \in S$. Then :$x \ll y$ {{iff}} :$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in {\it Fin}\left({X}\right): x \preceq \sup A$ where :$\ll$ denotes the way below relation, :${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Continuous Midpoint-Convex Function is Convex

Let $f$ be a real function which is defined on a real interval $I$. Let $f$ be midpoint-convex and continuous on $I$. Then $f$ is convex.


Power Function on Strictly Positive Base is Continuous

Let $a \in \R_{>0}$.


Continuous Strictly Midpoint-Concave Function is Strictly Concave

Let $f$ be strictly midpoint-concave and continuous on $I$. Then $f$ is strictly concave.


Continuous Midpoint-Concave Function is Concave

Let $f$ be midpoint-concave and continuous on $I$. Then $f$ is concave.


Continuous Strictly Midpoint-Convex Function is Strictly Convex

Let $f$ be strictly midpoint-convex and continuous on $I$. Then $f$ is strictly convex.


Power Function on Strictly Positive Base is Convex

Let $a \in \R_{>0}$ be a strictly positive real number. Let $f: \R \to \R$ be the real function defined as: :$\map f x = a^x$ where $a^x$ denotes $a$ to the power of $x$. Then $f$ is convex.


Natural Logarithm as Derivative of Exponential at Zero

Let $\ln: \R_{>0}$ denote the real natural logarithm. Then: :$\displaystyle \forall x \in \R_{>0}: \ln x = \lim_{h \mathop \to 0} \frac {x^h - 1} h$


Product of Increasing Positive Functions is Increasing

Let $f$ and $g$ be real functions defined on an interval $I$. Let $f$ and $g$ be increasing and positive on $I$. Then the product $f g$ is increasing on $I$.


Derivative of Exponential Function/Proof 5/Lemma

:$\forall x \in \R : n \ge \left\lceil{\left\vert{x}\right\vert}\right\rceil \implies \left\langle{\dfrac n {n + x} \left({1 + \dfrac x n}\right)^n}\right\rangle$ is increasing.


Change of Index Variable of Product

:$\displaystyle \prod_{R \left({i}\right)} a_i = \prod_{R \left({j}\right)} a_j$


Translation of Index Variable of Summation/Corollary

:$\displaystyle \sum_{j \mathop = m}^n a_j = \sum_{j \mathop = m + c}^{n + c} a_{j - c}$ where $c$ is an integer constant which is not dependent upon $j$.


Permutation of Indices of Product

:$\displaystyle \prod_{\map R j} a_j = \prod_{\map R {\map \pi j} } a_{\map \pi j}$


Translation of Index Variable of Product

:$\displaystyle \prod_{R \left({j}\right)} a_j = \prod_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \prod_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where: :$\displaystyle \prod_{R \left({j}\right)} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $R \left({j}\right)$ :$c$ is an integer constant which is not dependent upon $j$.


Exchange of Order of Product

:$\displaystyle \prod_{R \left({i}\right)} \prod_{S \left({j}\right)} a_{i j} = \prod_{S \left({j}\right)} \prod_{R \left({i}\right)} a_{i j}$


Product of Products over Overlapping Domains

:$\displaystyle \prod_{R \left({j}\right)} a_j \prod_{S \left({j}\right)} a_j = \left({\prod_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j}\right) \left({\prod_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j}\right)$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.


Topology is Locally Compact iff Ordered Set of Topology is Continuous

Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be the ordered set where $\preceq$ is the inclusion relation. Then :$(1): \quad T$ is locally compact implies $L$ is continuous :$(2): \quad T$ is $T_3$ space and $L$ is continuous implies $T$ is locally compact


Sum of Sequence of Power by Index

:$\displaystyle \sum_{j \mathop = 0}^n j x^j = \frac {n x^{n + 2} - \paren {n + 1} x^{n + 1} + x} {\paren {x - 1}^2}$ for $x \ne 1$.


Ordering of Series of Ordered Sequences

Let $\sequence {a_n}$ and $\sequence {b_n}$ be two sequences. Let $\displaystyle \sum_{n \mathop = 1}^{\infty} a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be convergent series. For each $n \in \N$, let $a_n < b_n$. Then: :$\displaystyle \sum_{n \mathop = 0}^\infty a_n < \displaystyle \sum_{n \mathop = 0}^\infty b_n$


Reciprocal of Real Exponential

:$\dfrac 1 {\map \exp x} = \map \exp {-x}$


Telescoping Series/Example 2

Let $\left \langle {b_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k - 1}$ Then: :$\displaystyle \sum_{k \mathop = m}^n a_k = b_n - b_{m - 1}$


Repunit Integer as Product of Base - 1 by Increasing Digit Integer

{{begin-eqn}} {{eqn | l = 9 \times 1 + 2       | r = 11 }} {{eqn | l = 9 \times 12 + 3       | r = 111 }} {{eqn | l = 9 \times 123 + 4       | r = 1111 }} {{eqn | l = 9 \times 1234 + 5       | r = 11111 }} {{eqn | l = 9 \times 12345 + 6       | r = 11111 }} {{eqn | o = \ldots }} {{end-eqn}} That is: :$\displaystyle 9 \sum_{j \mathop = 0}^n \left({n - j}\right)10^j + n + 1 = \sum_{j \mathop = 0}^n 10^j$


Repunit Integer as Product of Base - 1 by Increasing Digit Integer/General Result

:$\displaystyle \paren {b - 1} \sum_{j \mathop = 0}^n \paren {n - j}b^j + n + 1 = \sum_{j \mathop = 0}^n b^j$


Exponential of Real Number is Strictly Positive/Proof 5/Lemma

:$\forall x \in \R: \exp x \ne 0$


Product of Products

Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers. Let $\displaystyle \prod_{R \paren i} x_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \prod_{R \paren i} \paren {b_i c_i} = \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$


Summation of General Logarithms

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \prod_{R \left({i}\right)} a_i$ denote a product over $R$. Let the fiber of truth of $R$ be finite. Then: :$\displaystyle \log_b \left({\prod_{R \left({i}\right)} a_i}\right) = \sum_{R \left({i}\right)} \log_b a_i$


Way Below Closure is Directed in Bounded Below Join Semilattice

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $x \in S$. Then :$x^\ll$ is directed.


Bottom is Way Below Any Element

Let $\left({S, \preceq}\right)$ be a bounded below ordered set. Let $x \in S$. Then :$\bot \ll x$ where $\bot$ denotes the smallest element in $S$.


Equivalence of Definitions of Real Exponential Function/Proof 2

The following definitions of the exponential function are equivalent.


Product to n of Product to Index

:$\displaystyle \prod_{i \mathop = 0}^n \prod_{j \mathop = 0}^i a_i a_j = \prod_{i \mathop = 0}^n {a_i}^{n + 2}$


Bernoulli's Inequality/Corollary/General Result

For all $n \in \Z_{\ge 0}$: :$\displaystyle \prod_{j \mathop = 1}^n \left({1 - a_j}\right) \ge 1 - \sum_{j \mathop = 1}^n a^j$ where $0 < a_j < 1$ for all $j$.


Product of Sequence of 1 minus Reciprocal of Squares

For all $n \in \Z_{\ge 1}$: :$\displaystyle \prod_{j \mathop = 2}^n \paren {1 - \dfrac 1 {j^2} } = \dfrac {n + 1} {2 n}$


Product of Summations

:$\displaystyle \prod_{j \mathop = 1}^n \sum_{i \mathop = 1}^m a_{i j} = \sum_{1 \mathop \le i_1, \mathop \ldots \mathop , i_n \le m} a_{i_1 1} \cdots a_{i_n n}$


Summation of Powers over Product of Differences

:$\displaystyle \sum_{j \mathop = 1}^n \begin{pmatrix} {\dfrac { {x_j}^r} {\displaystyle \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \left({x_j - x_k}\right)} } \end{pmatrix} = \begin{cases} 0 & : 0 \le r < n - 1 \\ 1 & : r = n - 1 \\ \displaystyle \sum_{j \mathop = 1}^n x_j & : r = n \end{cases}$


Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r

:$\displaystyle \sum_{k \mathop = 1}^n \left({\dfrac {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne m} } \left({x + k - r}\right)} {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne k} } \left({k - r}\right)} }\right) = 1$ where $1 \le m \le n$ and $x$ is arbitrary.


Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r/Example

:$\dfrac {x \left({x - 2}\right) \left({x - 3}\right)} {\left({-1}\right) \left({-2}\right) \left({-3}\right)} + \dfrac {\left({x + 1}\right) \left({x - 1}\right) \left({x - 2}\right)} {\left({1}\right) \left({-1}\right) \left({-2}\right)} + \dfrac {\left({x + 2}\right) x \left({x - 1}\right)} {\left({2}\right) \left({1}\right) \left({-1}\right)} + \dfrac {\left({x + 3}\right) \left({x + 1}\right) x} {\left({3}\right) \left({2}\right) \left({1}\right)} = 1$


Inverse of Cauchy Matrix

Let $C_n$ be the square Cauchy matrix of order $n$: :$C_n = \begin{bmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{bmatrix}$ Then its inverse $C_n^{-1} = \sqbrk b_n$ can be specified as: :$\begin{bmatrix} b_{ij} \end{bmatrix} = \begin{bmatrix} \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {x_j + y_k} \paren {x_k + y_i} } {\displaystyle \paren {x_j + y_i} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {y_i - y_k} } } \end{bmatrix}$


Sum of Elements in Inverse of Cauchy Matrix

Let $C_n$ be the Cauchy matrix of order $n$ given by: :$C_n = \begin{bmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2 } & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2 } & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_m + y_1} & \dfrac 1 {x_m + y_2 } & \cdots & \dfrac 1 {x_m + y_n} \\ \end{bmatrix}$ Let $C_n^{-1}$ be its inverse, from Inverse of Cauchy Matrix: :$b_{i j} = \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {x_j + y_k} \paren {x_k + y_i} } {\displaystyle \paren {x_j + y_i} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {y_i - x_k} } }$ The sum of all the elements of $C_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = \sum_{k \mathop = 1}^n x_k + \sum_{k \mathop = 1}^n y_k$


Hilbert Matrix is Cauchy Matrix

A Hilbert matrix is a special case of a Cauchy matrix.


Inverse of Hilbert Matrix

Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Then its inverse $H_n^{-1} = \sqbrk b_n$ can be specified as: :$\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\paren {-1}^{i + j} \paren {i + n - 1}! \paren {j + n - 1}!} {\paren {\paren {i - 1}!}^2 \paren {\paren {j - 1}!}^2 \paren {n - j}! \paren {n - i}! \paren {i + j - 1} } \end{bmatrix}$


Sum of Elements in Inverse of Hilbert Matrix

Let $H_n$ be the Hilbert matrix of order $n$: :$\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$ Consider its inverse $H_n^{-1}$. All the elements of $H_n^{-1}$ are integers. The sum of all the elements $b_{i j}$ of $H_n^{-1}$ is: :$\displaystyle \sum_{1 \mathop \le i, \ j \mathop \le n} b_{i j} = n^2$


Suprema of two Real Sets

Let $S$ and $T$ be real sets. Let $S$ and $T$ admit suprema. Then: : $\sup S \le \sup T \iff \forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s < t + \epsilon$


Sequential Continuity is Equivalent to Continuity in the Reals

Let $A \subseteq \R$ be a subset of the real numbers. Let $c \in A$. Let $f : A \to \R$ be a real function. Then $f$ is continuous at $c$ {{iff}}: :for each sequence $\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.


Rising Factorial as Quotient of Factorials

:$x^{\overline n} = \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!} = \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$


Equivalent Definition for Alternating Bilinear Mapping

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$ with the property that $\operatorname{Char}\left({R}\right) \neq 2$. Then the following definitions for alternating bilinear maps are equivalent: * $\oplus$ is an alternating bilinear map {{iff}} for all $a \in A_R$, $a \oplus a = 0$. * $\oplus$ is an alternating bilinear map {{iff}} for all $a, b \in A_R$, $a \oplus b + b \oplus a = 0$.


Way Below if Between is Compact Set in Ordered Set of Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $L = \left({\tau, \preceq}\right)$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$ Let $x, y \in \tau$ such that :$\exists H \subseteq S: x \subseteq H \subseteq y \land H$ is compact Then :$x \ll y$


Locally Compact iff Open Neighborhood contains Compact Set

Let $T = \struct{S, \tau}$ be a topological space. Then :$T$ is locally compact {{iff}} :$\forall U \in \tau, x \in U: \exists K \subseteq S: x \in K^\circ \land K \subseteq U \land K$ is compact where $K^\circ$ denotes the interior of $K$.


Cauchy-Binet Formula

Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf B$ be an $n \times m$ matrix. Let $1 \le j_1, j_2, \ldots, j_m \le n$. Let $\mathbf A_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$. Let $\mathbf B_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf B$. Then: :$\displaystyle \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$ where $\det$ denotes the determinant.


Auxiliary Relation is Congruent

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R$ be relation on $S$ satisfying conditions $(ii)$ and $(iii)$ of auxiliary relation. Then :$\forall x, y, z, u \in S: \left({x, z}\right) \in \mathcal R \land \left({y, u}\right) \in \mathcal R \implies \left({x \vee y, z \vee u}\right) \in \mathcal R$


Cauchy-Binet Formula/Example/m equals 1

Let $\mathbf A = \left[{a}\right]_{1 n}$ be a row matrix with $n$ columns. and $\mathbf B = \left[{b}\right]_{n 1}$ be a column matrix with $n$ rows. Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$. Then: :$\displaystyle \det \left({\mathbf A \mathbf B}\right) = \sum_{j \mathop = 1}^n a_j b_j$ where: :$a_j$ is element $a_{1 j}$ of $\mathbf A$ :$b_j$ is element $b_{j 1}$ of $\mathbf B$.


Preceding is Auxiliary Relation

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Then :$\preceq$ is auxiliary relation.


Cauchy-Binet Formula/Example/Matrix by Transpose

Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf A^\intercal$ be the transpose $\mathbf A$. Let $1 \le j_1, j_2, \ldots, j_m \le n$. Let $\mathbf A_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of columns $j_1, j_2, \ldots, j_m$ of $\mathbf A$. Let $\mathbf A^\intercal_{j_1 j_2 \ldots j_m}$ denote the $m \times m$ matrix consisting of rows $j_1, j_2, \ldots, j_m$ of $\mathbf A^\intercal$. Then: :$\displaystyle \det \left({\mathbf A \mathbf A^\intercal}\right) = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \left({\det \left({\mathbf A_{j_1 j_2 \ldots j_m} }\right)}\right)^2$ where $\det$ denotes the determinant.


Preceding is Top in Ordered Set of Auxiliary Relations

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let ${\it Aux}\left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({ {\it Aux}\left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{{\it Aux}\left({L}\right) \times {\it Aux}\left({L}\right)}$ Then :$\preceq \mathop = \top_P$ where $\top_P$ denotes the greatest element in $P$.


Bottom Relation is Auxiliary Relation

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $B = \left\{ {\left({\bot, x}\right): x \in S}\right\}$ where $\bot$ denotes the smallest element in $L$. Then :$B$ is auxiliary relation.


Bottom Relation is Bottom in Ordered Set of Auxiliary Relations

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let ${\it Aux}\left({L}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({ {\it Aux}\left({L}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{{\it Aux}\left({L}\right) \times {\it Aux}\left({L}\right)}$ Let $B = \left\{ {\left({\bot_L, x}\right): x \in S}\right\}$ where $\bot_L$ denotes smallest element in $L$. Then :$B \mathop = \bot_P$


Inverse of Conditional is Converse of Contrapositive

Let $p \implies q$ be a conditional. Then the inverse of $p \implies q$ is the converse of its contrapositive.


Inverse of Conditional is Contrapositive of Converse

Let $p \implies q$ be a conditional. Then the inverse of $p \implies q$ is the contrapositive of its converse.


Converse of Conditional is Contrapositive of Inverse

Let $p \implies q$ be a conditional. Then the converse of $p \implies q$ is the contrapositive of its inverse.


Converse of Conditional is Inverse of Contrapositive

Let $p \implies q$ be a conditional. Then the converse of $p \implies q$ is the inverse of its contrapositive.


Intersection of Auxiliary Relations is Auxiliary Relation

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal F$ be a non-empty set of auxiliary relations on $S$. Then :$\bigcap \mathcal F$ is auxiliary relation.


Cauchy-Binet Formula/Example/m greater than n

Let $\mathbf A$ be an $m \times n$ matrix. Let $\mathbf B$ be an $n \times m$ matrix. Let $m > n$. Then: :$\displaystyle \det \left({\mathbf A \mathbf B}\right) = 0$


Krattenthaler's Identity

:$\begin{vmatrix} \left({x + q_2}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + q_3}\right) & \left({x + p_1}\right) \left({x + p_2}\right) \\ \left({y + q_2}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + q_3}\right) & \left({y + p_1}\right) \left({y + p_2}\right) \\ \left({z + q_2}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + q_3}\right) & \left({z + p_1}\right) \left({z + p_2}\right) \end{vmatrix} = \left({x - y}\right) \left({x - z}\right) \left({y - z}\right) \left({p_1 - q_2}\right) \left({p_1 - q_3}\right) \left({p_2 - q_3}\right)$ where $\left\vert{\, \cdot \,}\right\vert$ denotes determinant.


Infima of two Real Sets

Let $S$ and $T$ be sets of real numbers. Let $S$ and $T$ admit infima. Then: : $\inf S \ge \inf T \iff \forall \epsilon \in \R_{>0}: \forall s \in S: \exists t \in T: s + \epsilon > t$


Floor Function is Integer

Let $x$ be a real number. Then the floor function of $x$ is an integer: :$\floor x \in \Z$


Floor of Number plus Integer

:$\forall n \in \Z: \floor x + n = \floor {x + n}$


Floor is between Number and One Less

:$x - 1 < \left\lfloor{x}\right\rfloor \le x$ where $\left\lfloor{x}\right\rfloor$ is the floor of $x$.


Equivalence of Definitions of Floor Function

Let $x$ be a real number. {{TFAE|def = Floor Function}}


Properties of Floor Function

This page gathers together some basic propeties of the floor function.


Properties of Ceiling Function

This page gathers together some basic propeties of the ceiling function.


Ceiling Function is Integer

Let $x$ be a real number. Then the ceiling function of $x$ is an integer: :$\ceiling x \in \Z$


Real Number is Integer iff equals Ceiling

:$x = \ceiling x \iff x \in \Z$


Ceiling Function is Idempotent

:$\ceiling {\ceiling x} = \ceiling x$


Way Below Relation is Auxiliary Relation

Lrt $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Then :$\ll$ is auxiliary relation where $\ll$ denotes the way below relation.


Number is between Ceiling and One Less

:$\ceiling x - 1 < x \le \ceiling x$


Ceiling is between Number and One More

:$x \le \ceiling x < x + 1$ where $\ceiling x$ is the ceiling of $x$.


Ceiling of Number plus Integer

:$\forall n \in \Z: \left \lceil {x} \right \rceil + n = \left \lceil {x + n} \right \rceil$


Relation Segment of Auxiliary Relation is Ideal

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \in S$. Then :$x^R$ is ideal in $L$ where $x^R$ denotes the $R$-segment of $x$.


Quotient of Modulo Operation with Modulus

Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y \ne 0$. Then: :$0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor = \dfrac {x \bmod y} y < 1$


Range of Modulo Operation for Positive Modulus

Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$. Let $y > 0$. Then: :$0 \le x \bmod y < y$


Range of Modulo Operation for Negative Modulus

Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$. Let $y < 0$. Then: :$0 \ge x \bmod y > y$


Zero is Integer Multiple of Zero

Zero is an integer multiple of zero.


Modulo Operation/Examples/18 mod 3

:$18 \bmod 3 = 0$


Number is Divisor iff Modulo is Zero

Let $x, y \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation: :$x \bmod y := \begin {cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end {cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$. Then $x \bmod y = 0$ {{iff}} $x$ is an integer multiple of $y$.


Number less than Integer iff Floor less than Integer

:$\left \lfloor{x}\right \rfloor < n \iff x < n$


Number not less than Integer iff Floor not less than Integer

:$x \ge n \iff \left \lfloor{x}\right \rfloor \ge n$


Number not greater than Integer iff Ceiling not greater than Integer

:$\left \lceil{x}\right \rceil \le n \iff x \le n$


Number greater than Integer iff Ceiling greater than Integer

:$\left \lceil{x}\right \rceil > n \iff x > n$


Integer equals Floor iff between Number and One Less

:$\floor x = n \iff x - 1 < n \le x$


Integer equals Floor iff Number between Integer and One More

:$\floor x = n \iff n \le x < n + 1$


Integer equals Ceiling iff between Number and One More

:$\left \lceil{x}\right \rceil = n \iff x \le n < x + 1$


Integer equals Ceiling iff Number between Integer and One Less

:$\ceiling x = n \iff n - 1 < x \le n$


Floor of Root of Floor equals Floor of Root

:$\displaystyle \floor {\sqrt {\floor x} } = \floor {\sqrt x}$


Ceiling of Root of Ceiling equals Ceiling of Root

:$\displaystyle \ceiling {\sqrt {\ceiling x} } = \ceiling {\sqrt x}$


Congruence by Divisor of Modulus/Integer Modulus

Let $r, s \in \Z$ be integers. Let $a, b \in \Z$ such that $a$ is congruent modulo $r s$ to $b$, that is: :$a \equiv b \pmod {r s}$ Then: :$a \equiv b \pmod r$ and: :$a \equiv b \pmod s$


Modulo Addition is Well-Defined/Real Modulus

Let $z \in \R$ be a real number. Let: :$a \equiv b \pmod z$ and: :$x \equiv y \pmod z$ where $a, b, x, y \in \R$. Then: : $a + x \equiv b + y \pmod z$


Congruence by Product of Moduli/Real Modulus

Let $a, b, z \in \R$. Let $a \equiv b \pmod z$ denote that $a$ is congruent to $b$ modulo $z$. Then $\forall y \in \R, y \ne 0$: :$a \equiv b \pmod z \iff y a \equiv y b \pmod {y z}$


Segment of Auxiliary Relation is Subset of Lower Closure

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be auxiliary relation on $S$. Let $x \in S$. Then :$x^R \subseteq x^\preceq$ where :$x^R$ denotes the $R$-segment of $x$, :$x^\preceq$ denotes the lower closure of $x$.


Preceding implies Inclusion of Segments of Auxiliary Relation

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $R$ be an auxiliary relation on $S$. Let $x, y \in S$ such that :$x \preceq y$ Then :$x^R \subseteq y^R$ where $x^R$ denotes the $R$-segment of $x$.


Characteristic Function of Square-Free Integers is Multiplicative

Let $S \subseteq \Z$ be the set of positive integers defined as: :$S := \set {n \in \Z: \forall k \in \Z_{>1}: k^2 \nmid n}$ That is, let $S$ be the set of all square-free positive integers. Let $\chi_S: \N \to \Z$ denote the characteristic function of $S$: :$\forall n \in \Z: \map {\chi_S} n = \sqbrk {n \in S}$ where $\sqbrk {n \in S}$ is Iverson's convention. Then $\chi_S$ is multiplicative.


Segment of Auxiliary Relation Mapping is Increasing

Let $R = \left({S, \preceq}\right)$ be an ordered set. Let ${\it Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ {\it Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{ {\it Ids}\left({R}\right) \times {\it Ids}\left({R}\right)}$ Let $r$ be an auxiliary relation on $S$. Let $f: S \to {\it Ids}\left({R}\right)$ be a mapping such that :$\forall x \in S: f\left({x}\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then :$f$ is increasing mapping.


Product of Multiplicative Functions is Multiplicative

Let $f: \N \to \C$ and $g: \N \to \C$ be multiplicative functions. Then their pointwise product: :$f \times g: \Z \to \Z: \forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ is also multiplicative.


Segment of Auxiliary Relation Mapping is Element of Increasing Mappings Satisfying Inclusion in Lower Closure

Let $R = \left({S, \preceq}\right)$ be an ordered set. Let $\mathit{Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ \mathit{Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{\mathit{Ids}\left({R}\right) \times \mathit{Ids}\left({R}\right)}$. Let $r$ be an auxiliary relation on $S$. Let $M = \left({F, \preccurlyeq}\right)$ be the ordered set of increasing mappings $g$ satisfying $\forall x \in S: g\left({x}\right) \subseteq x^\preceq$. Let $f: S \to \mathit{Ids}\left({R}\right)$ be a mapping such that: :$\forall x \in S: f\left({x}\right) = x^r$ where $x^r$ denotes the $r$-segment of $x$. Then: :$(1): \quad f \in F$ Let $h: S \to \mathit{Ids}\left({R}\right): x \mapsto x^\preceq$ Then: :$(2): \quad h \in F \land f \preccurlyeq h$ Let $k: S \to \mathit{Ids}\left({R}\right): x \mapsto \left\{ {\bot}\right\}$ where $\bot$ denotes the smallest element in $L$. Then: :$(3): \quad k \in F$ and $k \preccurlyeq f$


Singleton of Bottom is Ideal

Let $\struct {S, \preceq}$ be a bounded below ordered set. Then :$\set \bot$ is an ideal in $\struct {S, \preceq}$ where $\bot$ denotes the smallest element in $S$.


Rising Factorial in terms of Falling Factorial

:$x^{\overline n} = \left({x + n - 1}\right)^{\underline n}$


Rising Factorial in terms of Falling Factorial of Negative

:$x^{\overline k} = \paren {-1}^k \paren {-x}^{\underline k}$


Falling Factorial as Quotient of Factorials

:$x^{\underline n} = \dfrac {x!} {\paren {x - n}!} = \dfrac {\map \Gamma {x + 1} } {\map \Gamma {x - n + 1} }$


One to Integer Rising is Integer Factorial

:$1^{\overline n} = n!$


Number to Power of One Rising is Itself

:$x^{\overline 1} = x$


Integer to Power of Itself Falling is Factorial

:$n^{\underline n} = n!$


Number to Power of One Falling is Itself

:$x^{\underline 1} = x$


Number of Permutations of One Less

:${}^{n - 1} P_n = {}^n P_n$ where ${}^k P_n$ denotes the number of ordered selections of $k$ objects from $n$.


Element of Increasing Mappings Satisfying Inclusion in Lower Closure is Generated by Auxiliary Relation

Let $R = \left({S, \preceq}\right)$ be a bounded below join semilattice. Let $\mathit{Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ \mathit{Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{\mathit{Ids}\left({R}\right) \times \mathit{Ids}\left({R}\right)}$. Let $M = \left({F, \preccurlyeq}\right)$ be the ordered set of increasing mappings $g:S \to \mathit{Ids}\left({R}\right)$ satisfying $\forall x \in S: g\left({x}\right) \subseteq x^\preceq$. Let $f \in F$. Then :there exists an auxiliary relation $\mathcal R$ on $S$ such that ::$\forall x \in S:f\left({x}\right) = x^{\mathcal R}$ where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.


Binomial Coefficient with Two/Corollary

:$\forall n \in \N: \dbinom n 2 = T_{n - 1} = \dfrac {n \paren {n - 1} } 2$ where $T_n$ is the $n$th triangular number.


Increasing Mappings Satisfying Inclusion in Lower Closure is Isomorphic to Auxiliary Relations

Let $R = \left({S, \preceq}\right)$ be a bounded below join semilattice. Let $\mathit{Ids}\left({R}\right)$ be the set of all ideals in $R$. Let $L = \left({ \mathit{Ids}\left({R}\right), \precsim}\right)$ be an ordered set where $\precsim \mathop = \subseteq\restriction_{\mathit{Ids}\left({R}\right) \times \mathit{Ids}\left({R}\right)}$. Let $M = \left({F, \preccurlyeq}\right)$ be the ordered set of increasing mappings $g:S \to \mathit{Ids}\left({R}\right)$ satisfying $\forall x \in S: g\left({x}\right) \subseteq x^\preceq$. Let $\mathit{Aux}\left({R}\right)$ be the set of all auxiliary relations on $S$. Let $P = \left({ \mathit{Aux}\left({R}\right), \precsim'}\right)$ be an ordered set where $\precsim' \mathop = \subseteq\restriction_{\mathit{Aux}\left({R}\right) \times \mathit{Aux}\left({R}\right)}$. Then :there exists an order isomorphism between $P$ and $M$


Binomial Theorem/Abel's Generalisation

:$\displaystyle \left({x + y}\right)^n = \sum_k \binom n k x \left({x - k z}\right)^{k - 1} \left({y + k z}\right)^{n - k}$


Sum over k of r Choose m+k by s Choose n+k

Let $s \in \R, r \in \Z_{\ge 0}, m, n \in \Z$. Then: :$\displaystyle \sum_k \binom r {m + k} \binom s {n + k} = \binom {r + s} {r - m + n}$


Sum over k of r Choose k by s+k Choose n by -1^r-k

Let $s \in \R, r \in \Z_{\ge 0}, n \in \Z$. Then: :$\displaystyle \sum_k \binom r k \binom {s + k} n \left({-1}\right)^{r - k} = \binom s {n - r}$


Sum over k of r-k Choose m by s+k Choose n

Let $m, n, r, s \in \Z_{\ge 0}$ such that $n \ge s$. Then: :$\displaystyle \sum_{k \mathop = 0}^r \binom {r - k} m \binom {s + k} n = \binom {r + s + 1} {m + n + 1}$


Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk

Let $r, s, t \in \R, n \in \Z$. Then: :$\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \left({n - k}\right)} {n - k} \frac r {r - t k} = \binom {r + s - t n} n$


Sum over k of r Choose k by s Choose k by k

:$\displaystyle \sum_k \binom r k \binom s k k = \binom {r + s - 1} {r - 1} s$


Relation Segment is Increasing

Let $S$ be a set. Let $\RR, \QQ$ be relations on $S$ such that :$\RR \subseteq \QQ$ Let $x \in S$. Then :$x^\RR \subseteq x^\QQ$ where $x^\RR$ denotes the $\RR$-segment of $x$.


Sum over k of n+k Choose 2 k by 2 k Choose k by -1^k over k+1

:$\displaystyle \sum_k \binom {n + k} {2 k} \binom {2 k} k \frac {\paren {-1}^k} {k + 1} = \sqbrk {n = 0}$


Intersection of Ideals with Suprema Succeed Element equals Way Below Closure of Element

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $x \in S$. Then :$\displaystyle \bigcap \left\{ {I \in \mathit{Ids}: x \preceq \sup I}\right\} = x^\ll$ where $\mathit{Ids}$ denotes the set of all ideals in $L$.


Way Below Closure is Ideal in Bounded Below Join Semilattice

Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice. Let $x \in S$. Then :$x^\ll$ is ideal in $L$.


Sum over k of r Choose k by -1^r-k by Polynomial

Let $r \in \Z_{\ge 0}$. Then: :$\displaystyle \sum_k \binom r k \paren {-1}^{r - k} \map {P_r} k = r! \, b_r$ where: :$\map {P_r} k = b_0 + b_1 k + \cdots + b_r k^r$ is a polynomial in $k$ of degree $r$.


Sum over k of r Choose k by s-kt Choose r by -1^k

Let $r \in \Z_{\ge 0}$. Then: :$\displaystyle \sum_k \binom r k \binom {s - k t} r \left({-1}\right)^k = t^r$ where $\dbinom r k$ etc. are binomial coefficients.


Multinomial Coefficient expressed as Product of Binomial Coefficients

:$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m} = \dbinom {k_1 + k_2} {k_1} \dbinom {k_1 + k_2 + k_3} {k_1 + k_2} \cdots \dbinom {k_1 + k_2 + \cdots + k_m} {k_1 + k_2 + \cdots + k_{m - 1} }$ where: :$\dbinom {k_1 + k_2 + \cdots + k_m} {k_1, k_2, \ldots, k_m}$ denotes a multinomial coefficient :$\dbinom {k_1 + k_2} {k_1}$ etc. denotes binomial coefficients.


Universal Closures are Semantically Equivalent

Let $\mathbf A$ be a WFF of predicate logic. Let $\mathbf B, \mathbf B'$ be universal closures of $\mathbf A$. Then $\mathbf B$ and $\mathbf B'$ are semantically equivalent.


Meet-Continuous iff if Element Precedes Supremum of Directed Subset then Element equals Supremum of Meet of Element by Directed Subset

Let $L = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Then :$L$ is meet-continuous {{iff}} :$\forall x \in S$, directed subset $D$ of $S: x \preceq \sup D \implies x = \sup \set {x \wedge d: d \in D}$


Continuous Lattice is Meet-Continuous

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Then $L$ is meet-continuous.


Number to Power of Zero Rising is One

:$x^{\overline 0} = 1$


Number to Power of Zero Falling is One

:$x^{\underline 0} = 1$


Product of Number by its Falling Factorial

Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$. Then: :$x x^{\underline n} = x^{\underline {n + 1} } + n x^{\underline n}$


Product of Number by its Rising Factorial

Let $x^{\overline n}$ denote the $n$th rising factorial power of $x$. Then: :$x x^{\overline n} = x^{\overline {n + 1} } - n x^{\overline n}$


Value of Formula under Assignment Determined by Free Variables

Let $\mathbf A$ be a WFF of predicate logic. Let $\mathcal A$ be a structure for predicate logic. Let $\sigma, \sigma'$ be assignments for $\mathbf A$ in $\mathcal A$ such that: :For each free variable $x$ of $\mathbf A$, $\sigma \left({x}\right) = \sigma' \left({x}\right)$ Then: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma'}\right]$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$ is the value of $\mathbf A$ under $\sigma$.


First Inversion Formula for Stirling Numbers

For all $m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_k \left[{n \atop k}\right] \left\{ {k \atop m}\right\} \left({-1}\right)^{n - k} = \delta_{m n}$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$\displaystyle \left\{ {k \atop m}\right\}$ denotes a Stirling number of the second kind :$\delta_{m n}$ denotes the Kronecker delta.


Unsigned Stirling Number of the First Kind of 1

:$\displaystyle {1 \brack n} = \delta_{1 n}$


Signed Stirling Number of the First Kind of 1

:$s \left({1, n}\right) = \delta_{1 n}$


Stirling Number of the Second Kind of 1

:$\displaystyle {1 \brace n} = \delta_{1 n}$


Second Inversion Formula for Stirling Numbers

For all $m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_k \left\{ {n \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{n - k} = \delta_{m n}$ where: :$\displaystyle \left\{ {n \atop k}\right\}$ denotes a Stirling number of the second kind :$\displaystyle \left[{k \atop m}\right]$ denotes an unsigned Stirling number of the first kind :$\delta_{m n}$ denotes the Kronecker delta.


Zero Choose n

:$\dbinom 0 n = \delta_{0 n}$


Unsigned Stirling Number of the First Kind of 0

:$\displaystyle \left[{0 \atop n}\right] = \delta_{0 n}$


Particular Values of Signed Stirling Numbers of the First Kind

This page gathers together some particular values of signed Stirling numbers of the first kind.


Particular Values of Unsigned Stirling Numbers of the First Kind

This page gathers together some particular values of unsigned Stirling numbers of the first kind.


Particular Values of Stirling Numbers of the Second Kind

This page gathers together some particular values of Stirling numbers of the second kind.


Stirling Number of the Second Kind of 0

:$\displaystyle {0 \brace n} = \delta_{0 n}$


Unsigned Stirling Number of the First Kind of Number with Self

:$\displaystyle \left[{n \atop n}\right] = 1$


Unsigned Stirling Number of the First Kind of Number with Greater

Let $\displaystyle {n \brack k}$ denote an unsigned Stirling number of the first kind. Then: :$\displaystyle {n \brack k} = 0$


Signed Stirling Number of the First Kind of Number with Greater

Let $\map s {n, k}$ denote a signed Stirling number of the first kind. Then: :$\map s {n, k} = 0$


Stirling Number of Number with Greater

Let $n, k \in \Z_{\ge 0}$. Let $k > n$.


Stirling Number of the Second Kind of Number with Self

:$\displaystyle {n \brace n} = 1$


Value of Term under Assignment Determined by Variables

Let $\tau$ be a term of predicate logic. Let $\mathcal A$ be a structure for predicate logic. Let $\sigma, \sigma'$ be assignments for $\tau$ in $\mathcal A$ such that: :For each variable $x$ occurring in $\tau$, $\sigma \left({x}\right) = \sigma' \left({x}\right)$ Then: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma'}\right]$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$ is the value of $\tau$ under $\sigma$.


Relation between Signed and Unsigned Stirling Numbers of the First Kind

Let $m, n \in \Z_{\ge 0}$ be positive integers. Then: :$\displaystyle {n \brack m} = \paren {-1}^{n + m} \map s {n, m}$ where: :$\displaystyle {n \brack m}$ denotes an unsigned Stirling number of the first kind :$\map s {n, m}$ denotes a signed Stirling number of the first kind.


Unsigned Stirling Number of the First Kind of Number with Greater/Proof 1

Let $n, k \in \Z_{\ge 0}$ such that $k > n$. {{:Unsigned Stirling Number of the First Kind of Number with Greater}}


Unsigned Stirling Number of the First Kind of Number with Greater/Proof 2


Stirling Number of the Second Kind of Number with Greater/Proof 1

Let $n, k \in \Z_{\ge 0}$ such that $k > n$. {{:Stirling Number of the Second Kind of Number with Greater}}


Stirling Number of the Second Kind of Number with Greater/Proof 2


Signed Stirling Number of the First Kind of Number with Self

:$s \left({n, n}\right) = 1$


Unsigned Stirling Number of the First Kind of n with n-1

:$\displaystyle \left[{n \atop n - 1}\right] = \binom n 2$


Stirling Number of the Second Kind of n with n-1

:$\displaystyle {n \brace n - 1} = \binom n 2$


Signed Stirling Number of the First Kind of n with n-1

:$\map s {n, n - 1} = -\dbinom n 2$


Unsigned Stirling Number of the First Kind of n+1 with 0

:$\displaystyle {n + 1 \brack 0} = 0$


Stirling Number of the Second Kind of n+1 with 0

:$\displaystyle {n + 1 \brace 0} = 0$


Unsigned Stirling Number of the First Kind of n+1 with 1

:$\displaystyle \left[{n + 1 \atop 1}\right] = n!$


Stirling Number of the Second Kind of n+1 with 1

:$\displaystyle {n + 1 \brace 1} = 1$


Signed Stirling Number of the First Kind of n+1 with 1

:$s \left({n + 1, 1}\right) = \left({-1}\right)^n n!$


Sum over k of Unsigned Stirling Numbers of the First Kind of n with k by k choose m

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \left[{n \atop k}\right] \binom k m = \left[{n + 1 \atop m + 1}\right]$ where: :$\displaystyle \left[{n \atop k}\right]$ denotes an unsigned Stirling number of the first kind :$\dbinom k m$ denotes a binomial coefficient.


Semantically Equivalent Terms are Equal

Let $\tau_1, \tau_2$ be terms. Suppose that they are semantically equivalent with respect to the empty set. Then $\tau_1 = \tau_2$.


Down Mapping is Generated by Approximating Relation

Let $L = \left({S, \wedge, \preceq}\right)$ be a bounded below meet-continuous meet semilattice. Let $\mathit{Ids}\left({L}\right)$ be the set of all ideals in $L$. Let $I$ be an ideal in $L$. Let $f: S \to \mathit{Ids}\left({L}\right)$ be a mapping such that :$\forall x \in S: x \preceq \sup I \implies f\left({x}\right) = \left\{ {x \wedge i: i \in I}\right\}$ and :$\forall x \in S: x \npreceq \sup I \implies f\left({x}\right) = x^\preceq$ where $x^\preceq$ denotes the lower closure of $x$. Then :there exists an auxiliary approximating relation $\mathcal R$ on $S$ such that ::$\forall s \in S: f\left({s}\right) = s^{\mathcal R}$ where $s^{\mathcal R}$ denotes the $\mathcal R$-segment of $s$.


Substitution Instance of Term is Term

Let $\beta, \tau$ be terms of predicate logic. Let $x \in \operatorname {VAR}$ be a variable. Let $\map \beta {x \gets \tau}$ be the substitution instance of $\beta$ substituting $\tau$ for $x$. Then $\map \beta {x \gets \tau}$ is a term.


Substitution Theorem for Terms

Let $\beta, \tau$ be terms. Let $x \in \mathrm{VAR}$ be a variable. Let $\beta \left({x \gets \tau}\right)$ be the substitution instance of $\beta$ substituting $\tau$ for $x$. Let $\mathcal A$ be a structure for predicate logic. Let $\sigma$ be an assignment for $\beta$ and $\tau$. Suppose that: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right] = a$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$ is the value of $\tau$ under $\sigma$. Then: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\beta \left({x \gets \tau}\right) }\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\beta}\right) } \left[{\sigma + \left({x / a}\right)}\right]$ where $\sigma + \left({x / a}\right)$ is the extension of $\sigma$ by mapping $x$ to $a$.


Sum over k of Unsigned Stirling Numbers of the First Kind of n+1 with k+1 by k choose m by -1^k-m

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \left[{n + 1 \atop k + 1}\right] \binom k m \left({-1}\right)^{k - m} = \left[{n \atop m}\right]$ where: :$\displaystyle \left[{n + 1 \atop k + 1}\right]$ etc. denotes an unsigned Stirling number of the first kind :$\dbinom k m$ denotes a binomial coefficient.


Substitution Theorem for Well-Formed Formulas

Let $\mathbf A$ be a WFF of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\tau$ be a term of predicate logic which is freely substitutable for $x$ in $\mathbf A$. Let $\mathbf A \left({x \gets \tau}\right)$ be the substitution instance of $\mathbf A$ substituting $\tau$ for $x$. Let $\mathcal A$ be a structure for predicate logic. Let $\sigma$ be an assignment for $\mathbf A$ and $\tau$. Suppose that: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right] = a$ where $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau}\right) } \left[{\sigma}\right]$ is the value of $\tau$ under $\sigma$. Then: :$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A \left({x \gets \tau}\right) }\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma + \left({x / a}\right)}\right]$ where $\sigma + \left({x / a}\right)$ is the extension of $\sigma$ by mapping $x$ to $a$.


Sum over k of Stirling Numbers of the Second Kind of k with m by n choose k

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \left\{ {k \atop m}\right\} \binom n k = \left\{ {n + 1 \atop m + 1}\right\}$ where: :$\displaystyle \left\{ {k \atop m}\right\}$ denotes a Stirling number of the second kind :$\dbinom n k$ denotes a binomial coefficient.


Sum over k of Stirling Numbers of the Second Kind of k+1 with m+1 by n choose k by -1^k-m

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom n k \left({-1}\right)^{n - k} = \left\{ {n \atop m}\right\}$ where: :$\displaystyle \left\{ {k + 1 \atop m + 1}\right\}$ etc. denotes a Stirling number of the second kind :$\dbinom n k$ denotes a binomial coefficient.


Universal Instantiation/Informal Statement

Suppose we have a universal statement: :$\forall x: \map P x$ where $\forall$ is the universal quantifier and $\map P x$ is a propositional function. Then we can deduce: :$\map P {\mathbf a}$ where $\mathbf a$ is any arbitrary object we care to choose in the universe of discourse. In natural language: :''Suppose $P$ is true of everything in the universe of discourse.'' :''Let $\mathbf a$ be an element of the universe of discourse." :''Then $P$ is true of $\mathbf a$.''


Universal Instantiation/Model

Let $\map {\mathbf A} x$ be a WFF of predicate logic. Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$. Then $\forall x: \map {\mathbf A} x \implies \map {\mathbf A} \tau$ is a tautology.


Existential Generalisation/Informal Statement

:$P \left({\mathbf a}\right) \vdash \exists x: P \left({x}\right)$ Suppose we have the following: : We can find an arbitrary object $\mathbf a$ in our universe of discourse which has the property $P$. Then we may infer that: : there exists in that universe ''at least one'' object $x$ which has that property $P$. This is called the '''Rule of Existential Generalisation''' and often appears in a proof with its abbreviation '''EG'''.


Way Below is Approximating Relation

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice. Then $\ll$ is an approximating relation on $S$.


Intersection of Relation Segments of Approximating Relations equals Way Below Closure

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below meet-continuous lattice. Let $\mathit{App}\left({L}\right)$ be the set of all auxiliary approximating relations on $S$. Let $x \in S$. Then : $\displaystyle \bigcap \left\{ {x^{\mathcal R}: \mathcal R \in \mathit{App}\left({L}\right)}\right\} = x^\ll$


Limit of Monotone Real Function/Increasing

Let $f$ be a real function which is increasing and bounded above on the open interval $\openint a b$. Let the supremum of $f$ on $\openint a b$ be $L$. Then: :$\displaystyle \lim_{x \mathop \to b^-} \map f x = L$ where $\displaystyle \lim_{x \mathop \to b^-} \map f x$ is the limit of $f$ from the left at $b$.


Limit of Monotone Real Function/Decreasing

Let $f$ be a real function which is decreasing and bounded below on the open interval $\openint a b$. Let the infimum of $f$ on $\openint a b$ be $l$. Then: :$\displaystyle \lim_{x \mathop \to b^-} \map f x = l$ where $\displaystyle \lim_{x \mathop \to b^-} \map f x$ is the limit of $f$ from the left at $b$.


Sum over k of m choose k by -1^m-k by k to the n

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \binom m k \paren {-1}^{m - k} k^n = m! {n \brace m}$ where: :$\dbinom m k$ denotes a binomial coefficient :$\displaystyle {n \brace m}$ etc. denotes a Stirling number of the second kind :$m!$ denotes a factorial.


Sum over k of m-n choose m+k by m+n choose n+k by Stirling Number of the Second Kind of m+k with k

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \binom {m - n} {m + k} \binom {m + n} {n + k} {m + k \brace k} = {n \brack n - m}$ where: :$\dbinom {m - n} {m + k}$ etc. denote binomial coefficients :$\displaystyle {m + k \brace k}$ denotes a Stirling number of the second kind :$\displaystyle {n \brack n - m}$ denotes an unsigned Stirling number of the first kind.


Sum over k of m-n choose m+k by m+n choose n+k by Unsigned Stirling Number of the First Kind of m+k with k

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \binom {m - n} {m + k} \binom {m + n} {n + k} {m + k \brack k} = {n \brace n - m}$ where: :$\dbinom {m - n} {m + k}$ etc. denote binomial coefficients :$\displaystyle {m + k \brack k}$ denotes an unsigned Stirling number of the first kind :$\displaystyle {n \brace n - m}$ denotes a Stirling number of the second kind.


Sum over k of Stirling Number of the Second Kind of n+1 with k+1 by Unsigned Stirling Number of the First Kind of k with m by -1^k-m

Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k \left\{ {n + 1 \atop k + 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{k - m} = \binom n m$ where: :$\dbinom n m$ denotes a binomial coefficient :$\displaystyle \left[{k \atop m}\right]$ denotes an unsigned Stirling number of the first kind :$\displaystyle \left\{ {n + 1 \atop k + 1}\right\}$ denotes a Stirling number of the second kind.


Intersection of Applications of Down Mappings at Element equals Way Below Closure of Element

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below meet-continuous lattice. Let $\mathit{Ids}$ be the set of all ideals in $L$. Let for all $I \in \mathit{Ids}$: $m_I: S \to \mathit{Ids}$ be a mapping: :$\forall x \in S: x \preceq \sup I \implies m_I\left({x}\right) = \left\{ {x \wedge i: i \in I}\right\}$ and :$\forall x \in S: x \npreceq \sup I \implies m_I\left({x}\right) = x^\preceq$ where $x^\preceq$ denotes the lower closure of $x$. Let $x \in S$. Then :$\displaystyle \bigcap \left\{ {m_I \left({x}\right): I \in \mathit{Ids} }\right\} = x^\ll$ where $x^\ll$ denotes the way below closure of $x$.


Zero Choose Zero

:$\dbinom 0 0 = 1$


Intersection of Lower Closure of Element with Ideal equals Meet of Element and Ideal

Let $\struct {S, \preceq}$ be a meet semilattice. Let $I$ be an ideal in $\struct {S, \preceq}$. Let $x \in S$. Then: :$\paren {x^\preceq} \cap I = \set {x \wedge i: i \in I}$ where $x^\preceq$ denotes the lower closure of $x$.


Condition on Lower Coefficient for Binomial Coefficient to be Maximum

Let $n, k \in \Z_{\ge 0}$ be positive integers. Let $\dbinom n k$ denote the binomial coefficient of $n$ choose $k$. Then for a given value of $n$, the value of $k$ for which $\dbinom n k$ is a maximum is: :$k_\max = \floor {\dfrac n 2}$ and also: :$k_\max = \ceiling {\dfrac n 2}$


N Choose Negative Number is Zero

Let $n \in \Z$ be an integer. Let $k \in \Z_{<0}$ be a (strictly) negative integer. Then: :$\dbinom n k = 0$


Kummer's Theorem

Let $p$ be a prime number. Let $a, b \in \Z_{\ge 0}$. Let: :$p^n \divides \dbinom {a + b} b$ but :$p^{n + 1} \nmid \dbinom {a + b} b$ where: :$\divides$ denotes divisibility :$\nmid$ denotes non-divisibility :$\dbinom {a + b} b$ denotes a binomial coefficient. Then $n$ equals the number of carries that occur when $a$ is added to $b$ using the classical addition algorithm in base $p$.