Sum of Unitary Divisors of Integer

Let $n$ be an integer such that $n \ge 2$. Let $\map {\sigma^*} n$ be the sum of all positive unitary divisors of $n$. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ Then: :$\displaystyle \map {\sigma^*} n = \prod_{1 \mathop \le i \mathop \le r} \paren {1 + p_i^{k_i}}$


Range of Infinite Sequence may be Finite

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence. Then it is possible for the range of $\sequence {x_n}$ to be finite.


Largest Number not Expressible as Sum of Multiples of Coprime Integers

Let $a, b$ be coprime integers, each greater than $1$. Then the largest number not expressible as a sum of multiples of $a$ and $b$ is the number: :$a b - a - b = \paren {a - 1} \paren {b - 1} - 1$


Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space

Let $I := \closedint a b$ be a closed real interval. Let $\map \CC I$ be a space of continuous on closed interval real-valued functions. Let $\map {\CC^1} I$ be a space of continuously differentiable functions on closed interval $I$. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition of real-valued functions. Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of real-valued functions. Then $\struct {\map {\CC^1} I, +, \, \cdot \,}_\R$ is a vector space.


Fermat Quotient of 2 wrt p is Square iff p is 3 or 7/Generalization

The Fermat quotient of $2$ with respect to $p$: :$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$ is a perfect power {{iff}} $p = 3$ or $p = 7$.


Taxicab Norm is Norm

The taxicab norm is a norm on the real and complex numbers.


Multiplication by 2 over 3 in Egyptian Fractions

Let $\dfrac 1 n$ be an Egyptian fraction not equal to $\dfrac 2 3$. In order to multiply $\dfrac 1 n$ by $\dfrac 2 3$ and have it that $\dfrac 1 n \times \dfrac 2 3$ is also expressed in Egyptian form, we have: :$\dfrac 1 n \times \dfrac 2 3 = \dfrac 1 {2 n} + \dfrac 1 {6 n}$


Proper Fraction can be Expressed as Finite Sum of Unit Fractions

Let $\dfrac p q$ denote a proper fraction expressed in canonical form. Then it is always possible to express $\dfrac p q$ as the sum of a finite number of distinct unit fractions: {{begin-eqn}} {{eqn | l = \dfrac p q       | r = \sum_{\substack {1 \mathop \le k \mathop \le m \\ n_j \mathop \le n_{j + 1} } } \dfrac 1 {n_k}       | c =  }} {{eqn | r = \dfrac 1 {n_1} + \dfrac 1 {n_2} + \dotsb + \dfrac 1 {n_m}       | c =  }} {{end-eqn}}


Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm

Let $\dfrac p q$ denote a proper fraction expressed in canonical form. Let $\dfrac p q$ be expressed as the sum of a finite number of distinct unit fractions using Fibonacci's Greedy Algorithm. Then $\dfrac p q$ is expressed using no more than $p$ unit fractions.


Smallest n for which 3 over n produces 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient

Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$. The smallest $n$ for which $S$ consists of $3$ terms, where $2$ would be sufficient, is $25$.


Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function

Let $I = \closedint a b$ be a closed interval. Let $\struct {\map \CC I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$. Let $\map x t \in \map \CC I$ be a continuous real function. Let $\size {\, \cdot \,}$ be the absolute value. Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\norm {\, \cdot \,}_\infty$ is a norm over $\struct {\map \CC I, +, \, \cdot \,}_\R$.


Union with Disjoint Singleton is Dependent if Element Depends on Subset

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x \in S : x \notin A$. If $x$ depends on $A$ then $A \cup \set x$ is dependent


Element Depends on Independent Set iff Union with Singleton is Dependent

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X \in \mathscr I$. Let $x \in S : x \notin X$. Then: :$x \in \map \sigma X$ {{iff}} $X \cup \set x$ is dependent.


Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma

Let $A \in \mathscr I$ such that $A \subseteq X \cup \set x$. Then: :$\size A \le \size X$


Rank of Independent Subset Equals Cardinality

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho : \powerset S \to \Z$ be the rank function of $M$. Let $X \in \mathscr I$ Then: :$\map \rho X = \size X$


Generating Function for Lucas Numbers

Let $\map G z$ be the function defined as: :$\map G z = \dfrac {2 - z} {1 - z - z^2}$ Then $\map G z$ is a generating function for the Lucas numbers.


492 Cubed is Sum of 3 Positive Cubes in 13 Ways

The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways: {{begin-eqn}} {{eqn | l = 492^3       | r = 24^3 + 204^3 + 480^3 }} {{eqn | r = 48^3 + 85^3 + 491^3 }} {{eqn | r = 72^3 + 384^3 + 396^3 }} {{eqn | r = 113^3 + 264^3 + 463^3 }} {{eqn | r = 114^3 + 360^3 + 414^3 }} {{eqn | r = 149^3 + 336^3 + 427^3 }} {{eqn | r = 176^3 + 204^3 + 472^3 }} {{eqn | r = 190^3 + 279^3 + 449^3 }} {{eqn | r = 207^3 + 297^3 + 438^3 }} {{eqn | r = 226^3 + 332^3 + 414^3 }} {{eqn | r = 243^3 + 358^3 + 389^3 }} {{eqn | r = 246^3 + 328^3 + 410^3 }} {{eqn | r = 281^3 + 322^3 + 399^3 }} {{end-eqn}}


Maximum Area of Isosceles Triangle

Consider two line segments $A$ and $B$ of equal length $a$ which are required to be the legs of an isosceles triangle $T$. Then the area of $T$ is greatest when the apex of $T$ is a right angle. The area of $T$ in this situation is equal to $\dfrac {a^2} 2$.


Inscribing Equilateral Triangle inside Square with a Coincident Vertex

Let $\Box ABCD$ be a square. It is required that $\triangle DGH$ be an equilateral triangle inscribed within $\Box ABCD$ such that vertex $D$ of $\triangle DGH$ coincides with vertex $D$ of $\Box ABCD$.


Construction of Perpendicular using Rusty Compass

Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to construct a straight line at right angles to $AB$ from the endpoint $A$, without extending $AB$ past $A$.


Division of Straight Line into Equal Parts using Rusty Compass

Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to divide $AB$ into as many equal parts as required.


Construction of Regular Pentagon using Rusty Compass

Using a straightedge and rusty compass, it is possible to inscribe a regular pentagon inside a circle.


Sum to Infinity of 2x^2n over n by 2n Choose n

For $\cmod x < 1$: :$\displaystyle \frac {2 x \arcsin x} {\sqrt {1 - x^2} } = \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}$


Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x

:$\displaystyle \int_0^\pi \map \ln {a + b \cos x} \rd x = \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$


Arccosine in terms of Arctangent

:$\displaystyle \arccos x = 2 \map \arctan {\sqrt {\frac {1 - x} {1 + x} } }$


Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x

:$\displaystyle \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x = \frac 1 {\sqrt {a^2 - b^2} } \map \arccos {\frac b a}$


Definite Integral to Infinity of Exponential of -a x by Sine of b x over x

:$\displaystyle \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$


Definite Integral to Infinity of Sine of m x over Exponential of 2 Pi x minus One

:$\displaystyle \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x = \frac 1 4 \coth \frac m 2 - \frac 1 {2 m}$


Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x

:$\displaystyle \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$


Fourier Series/Logarithm of Sine of x over 0 to Pi

:$\displaystyle \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$


Subset of Set Difference iff Disjoint Set

Let $S, T$ be sets. Let $A \subseteq S$ Then: :$A \cap T = \varnothing \iff A \subseteq S \setminus T$ where: :$A \cap T$ denotes set intersection :$\varnothing$ denotes the empty set :$S \setminus T$ denotes set difference.


Set Difference of Doubleton and Singleton is Singleton

Let $x, y$ be distinct objects. Then: :$\set{x, y} \setminus \set x = \set y$


Egyptian Formula for Area of Quadrilateral

Let $\Box ABCD$ be a quadrilateral. Let the sides of $\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is opposite $c$ and $b$ is opposite $d$. Then the area of $\Box ABCD$ can be approximated by: :$\map \Area {\Box ABCD} \approx \dfrac {a + c} 2 \times \dfrac {b + d} 2$ The closer $\Box ABCD$ is to a rectangle, the better the approximation.


1 plus Perfect Power is not Power of 2

The equation: :$1 + a^n = 2^m$ has no solutions in the integers for $n, m > 1$. This is an elementary special case of Catalan's Conjecture.


Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx

For $n \in \N_{>0}$: :$\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \ \d x = -\frac \pi {4 n}$


Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma

Let $a, b \in S$. Let $\set a$ and $\set b$ be independent. Then $\set {a, b}$ is dependent {{iff}}: :$a \in \map \sigma {\set b}$ and :$b \in \map \sigma {\set a}$


Definite Integral to Infinity of Sine m x over x by x Squared plus a Squared

:$\displaystyle \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x = \frac \pi {2 a^2} \paren {1 - e^{-m a} }$


Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x

:$\displaystyle \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$


Definite Integral from 0 to Pi of Sec x by Logarithm of One plus b Cosine x over One plus a Cosine x

:$\displaystyle \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \frac 1 2 \paren {\paren {\arccos a}^2 - \paren {\arccos b}^2}$


Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared

:$\displaystyle \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x = \frac {\pi \paren {\size q - \size p} } 2$


Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x

:$\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \arctan \frac b p - \arctan \frac a p$


Definite Integral of Periodic Function

Let $f$ be a Darboux integrable periodic function with period $L$. Let $\alpha \in \R$ and $n \in \Z$. Then: :$\displaystyle \int_\alpha^{\alpha + n L} \map f x \d x = n \int_0^L \map f x \d x$


Independent Subset is Contained in Maximal Independent Subset

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $X \in \mathscr I$ such that $X \subseteq A$. Then: :$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$ where $\rho$ is the rank function on $M$.


Automorphic Numbers in Base 10

If leading zeroes are allowed, there are exactly $4$ $n$-digit automorphic numbers in base $10$: :$00 \dots 00$ :$00 \dots 01$ :$5^{2^{n - 1} } \pmod {10^n}$ :$6^{5^{n - 1} } \pmod {10^n}$


Seventeen Horses/General Problem 1

A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs. They are to be distributed in the ratio $\dfrac 1 a : \dfrac 1 b : \dfrac 1 c$. Let $\dfrac 1 a + \dfrac 1 b + \dfrac 1 c < 1$. Then there are $7$ possible values of $\tuple {n, a, b, c}$ such that the required shares are: :$\dfrac {n + 1} a, \dfrac {n + 1} b, \dfrac {n + 1} c$ These values are: :$\tuple {7, 2, 4, 8}, \tuple {11, 2, 4, 6}, \tuple {11, 2, 3, 12}, \tuple {17, 2, 3, 9}, \tuple {19, 2, 4, 5}, \tuple {23, 2, 3, 8}, \tuple {41, 2, 3, 7}$ leading to shares, respectively, of: :$\tuple {4, 2, 1}, \tuple {6, 3, 2}, \tuple {6, 4, 1}, \tuple {9, 6, 2}, \tuple {10, 5, 4}, \tuple {12, 8, 3}, \tuple {21, 14, 6}$


Seventeen Horses/General Problem 2

A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs. They are to be distributed in the ratio $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$. Let $t = \dfrac q r = \displaystyle \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in canonical form. Let $t \ne 1$. Then it is possible to achieve the required share by adding $s$ objects to the existing $n$ such that: :$s + q = r$ when $q = n$. This still works whether $q$ is positive or negative.


Sum of Sequence of Products of 3 Consecutive Reciprocals

:$\displaystyle \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }$


Sum of Sequence of Products of 3 Consecutive Reciprocals/Corollary

:$\displaystyle \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac 1 4$


Heronian Triangle whose Altitude and Sides are Consecutive Integers

There exists exactly one Heronian triangle one of whose altitudes and its sides are all consecutive integers. This is the Heronian triangle whose sides are $\tuple {13, 14, 15}$ and which has an altitude $12$.


Integer Heronian Triangle can be Scaled so Area equals Perimeter

Let $T_1$ be an integer Heronian triangle whose sides are $a$, $b$ and $c$. Then there exists a rational number $k$ such that the Heronian triangle $T_2$ whose sides are $k a$, $k b$ and $k c$ such that the perimeter of $T$ is equal to the area of $T$.


Heronian Triangle is Similar to Integer Heronian Triangle

Let $\triangle {ABC}$ be a Heronian triangle. Then there exists an integer Heronian triangle $\triangle {A'B'C'}$ such that $\triangle {ABC}$ and $\triangle {A'B'C'}$ are similar.


Triple with Sum and Product Equal

For $a, b, c \in \Z$, $a \le b \le c$, the solutions to the equation: :$a + b + c = a b c$ are: :$\tuple {1, 2, 3}$ :$\tuple {-3, -2, -1}$ and the trivial solution set: :$\set {\tuple {-z, 0, z}: z \in \N}$


Triple with Product Quadruple the Sum

Let $a, b, c \in \N$ such that $a \le b \le c$. Then the solutions to: :$a b c = 4 \paren {a + b + c}$ are: :$\tuple {0, 0, 0}, \tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$


Proper Integer Heronian Triangle whose Area is 24

There exists exactly one proper integer Heronian triangle whose area equals $24$. That is, the obtuse triangle whose sides are of length $4$, $13$ and $15$.


Semiperimeter of Integer Heronian Triangle is Composite

The semiperimeter of an integer Heronian triangle is always a composite number.


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Necessary Condition

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr B$ be the set of bases of the matroid on $M$. Then $\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}


Independent Subset is Base if Cardinality Equals Rank of Matroid

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $B \in \mathscr I$ such that: :$\size B = \map \rho S$ Then: :$B$ is a base of $M$.


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition

Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$ satisfying the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then $\mathscr B$ is the set of bases of a matroid on $S$.


Morley's Trisector Theorem

Let $\triangle ABC$ be a triangle. Let the internal angles of $\triangle ABC$ be trisected. Let the points where these angle trisectors first intersect be $D$, $E$ and $F$. :500px Then $\triangle EDF$ is equilateral.


Mean Number of Elements Fixed by Self-Map

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$. Let $S^S$ be the set of all mappings from $S$ to itself. Let $\map \mu n$ denote the arithmetic mean of the number of fixed points of all the mappings in $S^S$. Then: :$\map \mu n = 1$


Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient

Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$. Then $S$ consists of $3$ terms, where $2$ would be sufficient {{iff}} the following conditions hold: :$n \equiv 1 \pmod 6$ :$\exists d: d \divides n$ and $d \equiv 2 \pmod 3$


P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences form Vector Space

Let $\ell^p$ be the p-sequence space. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition on the ring of sequences. Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences. Then $\struct {\ell^p, +, \, \cdot \,}_\R$ is a vector space.


Orthogonal Latin Squares of Order 6 do not Exist

Two orthogonal Latin squares of order $6$ do not exist.


Divisor of Product

Let $a, b, c \in \Z$ be integers. Let the symbol $\divides$ denote the divisibility relation. Let $a \divides b c$. Then there exist integers $r, s$ such that: :$a = r s$, where $r \divides b$ and $s \divides c$.


Equivalence of Definitions of Matroid

Let $M = \struct {S, \mathscr I}$ be an independence system. {{TFAE|def=Matroid}}


P-adic Norm is Well Defined

P-adic norm $\norm {\, \cdot \,}_p$ is well defined.


Equivalence of Definitions of Matroid/Definition 1 implies Definition 2

Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}


Equivalence of Definitions of Matroid/Definition 2 implies Definition 3

Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{ \size {V \cup Z} = \size U} }} {{end-axiom}}


Equivalence of Definitions of Matroid/Definition 3 implies Definition 1

Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}


Equivalence of Definitions of Matroid/Definition 1 implies Definition 4

Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}}


Equivalence of Definitions of Matroid/Definition 4 implies Definition 1

Let $M = \struct {S, \mathscr I}$ be an independence system. Let $M$ also satisfy: {{begin-axiom}} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}} Then $M$ satisfies: {{begin-axiom}} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}


Cardinality of Set Difference

Let $S$ and $T$ be sets such that $T$ is finite. Then: :$\card {S \setminus T} = \card S - \card {S \cap T}$ where $\card S$ denotes the cardinality of $S$.


Set Difference and Intersection are Disjoint

Let $S$ and $T$ be sets. Then:  :$S \setminus T$ and $S \cap T$ are disjoint  where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection.


Straight Line has Zero Curvature

A straight lines has zero curvature.


Partial Differential Equation of Spheres in 3-Space

The set of spheres in real Cartesian $3$-dimensional space can be described by the system of partial differential equations: :$\dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_x} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$ and if the spheres are expected to be real: :$z_{xx} z_{yy} > z_{xy}$


Fermat's Right Triangle Theorem

$x^4 + y^4 = z^2$ has no solutions in the (strictly) positive integers.


Taylor's Theorem/One Variable with Two Functions

Let $f$ and $g$ be real functions satisfying following conditions: :$(1): \quad f$ is $n + 1$ times differentiable on the open interval $\openint a x$ :$(2): \quad f$ is of differentiability class $C^n$ on the closed interval $\closedint a x$ :$(3): \quad g$ is $k + 1$ times differentiable on the open interval $\openint a x$ :$(4): \quad g$ is of differentiability class $C^k$ on the closed interval $\closedint a x$ :$(5): \quad \map {g^{\paren {k + 1}}} t \ne 0$ for any $t \in \openint a x$ Then the following equation holds for some real number $\xi \in \openint a x$: :$\dfrac {\map {f^{\paren {n + 1} } } \xi /n!} {\map {g^{\paren {k + 1} } } \xi /k!} \paren {x - \xi}^{n - k} = \dfrac {\map f x - \map f a - \map {f'} a \paren {x - a} - \dfrac {\map {f''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n} {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k}$ or equivalently: {{begin-eqn}} {{eqn | l = \map f x       | r = \map f a + \map {f'} a \paren {x - a} + \dfrac {\map {f''} a} {2!} \paren {x - a}^2 + \dotsb + \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n + R_n }} {{eqn | l = R_n       | r = \dfrac {\map {f^{\paren {n + 1} } } \xi / n!} {\map {g^{\paren {k + 1} } } \xi / k!} \paren {x - \xi}^{n - k} \paren {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g''} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k} }} {{end-eqn}}


Set of Even Integers is Countably Infinite

Let $\Bbb E$ be the set of even integers. Then $\Bbb E$ is countably infinite.


Basis Expansion of Rational Number

Let $x$ be a rational number. {{WIP|Time has caught up with me, I'll continue this later. I'm late for work.}} q2llusuyprk7567f7o3l3jnktxipxcq


Integration by Parts/Definite Integral

:$\displaystyle \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$


Integration by Parts/Primitive

:$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$ on $\closedint a b$.


Integration by Substitution/Primitive

The primitive of $f$ can be evaluated by: :$\ds \int \map f x \rd x = \int \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$ where $x = \map \phi u$.


Product Formula for Norms on Non-zero Rationals/Lemma

Let $z \in \Z_{\ne 0}$. Then the following infinite product converges: :$\size z \times \displaystyle\prod_{p \mathop \in \Bbb P}^{} \norm z_p = 1$


P-adic Open Ball is Instance of Open Ball of a Norm

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $B \subseteq \Q_p$. Then:  :$B$ is an open ball in $p$-adic numbers with radius $\epsilon$ and centre $a$ {{iff}}: :$B$ is an open ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$ . That is, the definition of an open ball in $p$-adic numbers is a specific instance of the general definition of an open ball in a normed division ring.


P-adic Closed Ball is Instance of Closed Ball of a Norm

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $B \subseteq \Q_p$. Then:  :$B$ is a closed ball in $p$-adic numbers with radius $\epsilon$ and centre $a$ {{iff}}: :$B$ is a closed ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a closed ball in a normed division ring.


P-adic Sphere is Instance of Sphere of a Norm

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $S \subseteq \Q_p$. Then:  :$S$ is a sphere in $p$-adic numbers with radius $\epsilon$ and centre $a$ {{iff}}: :$S$ is a sphere of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$  That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a sphere in a normed division ring.


Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers

Let $p$ be a prime number. Let $\Q_p$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\Q_p$. Let $\map {B_\epsilon} a$ denote the $\epsilon$-open ball of $a$ in $\Q_p$. Let $\map {S_\epsilon} a$ denote the $\epsilon$-sphere of $a$ in $\Q_p$. Then: :$\map {S_\epsilon} a = \map { {B_\epsilon}^-} a \setminus \map {B_\epsilon} a$


Set is Closed in Metric Space iff Closed in Induced Topological Space

Let $M = \struct {A, d}$ be a metric space. Let $\tau$ be the topology induced by the metric $d$. Let $F$ be a subset of $M$. Then: :$F$ is closed in $M$ {{iff}} $F$ is closed in $\struct {A, \tau}$


Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form

:$\displaystyle \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = -\arccos \frac x a + C$


Negative of Logarithm of x plus Root x squared minus a squared

Let $x \in \R: \size x > 1$. Let $x > 1$. Then: :$-\map \ln {x + \sqrt {x^2 - a^2} } = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}$


Supremum Norm on Vector Space of Real Matrices is Norm

Supremum Norm forms a norm on the vector space of real matrices.


Arccotangent Logarithmic Formulation

For any real number $x$: :$\arccot x = \dfrac 1 2 i \, \map \ln {\dfrac {1 + i x} {1 - i x} }$ where $\arccot x$ is the arccotangent and $i^2 = -1$.


Primitive of Reciprocal of x squared plus a squared/Arccotangent Form

:$\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$


Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form

:$\displaystyle \int \frac {\d x} {x \sqrt {x^2 + a^2} } = \frac 1 a \map \ln {\frac x {a + \sqrt {x^2 + a^2} } } + C$


Derivative of Hyperbolic Sine

:$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$


Open Ball is Open Set/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $\epsilon \in \R_{>0}$. Let $B_\epsilon \left({x}\right)$ be an open $\epsilon$-ball of $x$ in $M$. Then $B_\epsilon \left({x}\right)$ is an open set of $M$.


Derivative of Hyperbolic Cosine

:$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$


Derivative of Hyperbolic Tangent

:$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x$


Derivative of Hyperbolic Cotangent

:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x$


Derivative of Hyperbolic Secant

:$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$


Derivative of Hyperbolic Cosecant

:$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$


Derivative of Inverse Hyperbolic Sine Function

:$\map {\dfrac \d {\d x} } {\sinh^{-1} u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$


Derivative of Inverse Hyperbolic Cosine Function

:$\map {\dfrac \d {\d x} } {\cosh^{-1} u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$  where $u > 1$


Derivative of Inverse Hyperbolic Tangent Function

:$\map {\dfrac \d {\d x} } {\tanh^{-1} u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$  where $\size u < 1$


Derivative of Inverse Hyperbolic Cosecant Function

:$\map {\dfrac \d {\d x} } {\csch^{-1} u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$


Intersection of Closed Sets is Closed/Topology

Let $T = \struct {S, \tau}$ be a topological space. Then the intersection of an arbitrary number of closed sets of $T$ (either finitely or infinitely many) is itself closed.


Finite Union of Closed Sets is Closed/Topology

Let $T = \struct {S, \tau}$ be a topological space. Then the union of finitely many closed sets of $T$ is itself closed.


Closed Ball is Closed/Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $x \in A$. Let $\epsilon \in \R_{>0}$. Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon^-} x$ is a closed set of $M$.


Derivative of Sine of Function

:$\map {\dfrac \d {\d x} } {\sin u} = \cos u \dfrac {\d u} {\d x}$


Derivative of Secant of Function

:$\map {\dfrac \d {\d x} } {\sec u} = \sec u \tan u \dfrac {\d u} {\d x}$


Derivative of Cosecant of Function

:$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$


Derivative of Natural Logarithm of Function

:$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$


Derivative of Constant to Power of Function

:$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$


Derivative of Arcsine of Function

:$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$


Derivative of Arccosine of Function

:$\map {\dfrac \d {\d x} } {\arccos u} = -\dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$


Derivative of Arctangent of Function

:$\map {\dfrac \d {\d x} } {\arctan u} = \dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$


Derivative of Arccotangent of Function

:$\map {\dfrac \d {\d x} } {\arccot u} = -\dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$


Derivative of Arcsecant of Function

:$\map {\dfrac \d {\d x} } {\arcsec u} = \dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$


Derivative of Arccosecant of Function

:$\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$


Derivative of Odd Function is Even

Let $f$ be a differentiable real function such that $f$ is odd. Then its derivative $f'$ is an even function.


Form of Prime Sierpiński Number of the First Kind

Suppose $S_n = n^n + 1$ is a prime Sierpiński number of the first kind. Then: :$n = 2^{2^k}$ for some integer $k$.


Motion of Body Falling through Air

The motion of a body $B$ falling through air can be described using the following differential equation: :$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$ where: :$m$ denotes mass of $B$ :$y$ denotes the height of $B$ from an arbitrary reference :$t$ denotes time elapsed from an arbitrary reference :$g$ denotes the Acceleration Due to Gravity of $B$ :$k$ denotes the coefficient of resistive force exerted on $B$ by the air (assumed to be proportional to the speed of $B$)


Rationals are Everywhere Dense in Reals/Normed Vector Space

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers. Let $\Q$ be the set of rational numbers. Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$


Number of Parameters of Autoregressive Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ Then $M$ has $p + 2$ parameters. </onlyinclude>


Autoregressive Model is Special Case of Linear Filter Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$(1): \quad \tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ Then $M$ is a special case of a linear filter model. </onlyinclude>


Irrationals are Everywhere Dense in Reals/Topology

Let $T = \struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\R \setminus \Q$ be the set of irrational numbers. Then $\R \setminus \Q$ is everywhere dense in $T$.


Irrationals are Everywhere Dense in Reals/Normed Vector Space

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers. Let $\R \setminus \Q$ be the set of irrational numbers. Then $\R \setminus \Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$


Real Number Subtracted from Itself leaves Zero

Let $x \in \R$ be a real number. Then: :$x - x = 0$ where $x - x$ denotes the operation of real subtraction.


Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary

:$\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom

Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma

Let $B_1, B_2 \subseteq S$. Let $x \in B_1 \setminus B_2$. Let $y \in B_2 \setminus B_1$. Then: :$\paren{B_1 \setminus \set x} \cup \set y = \paren{B_1 \cup \set y} \setminus \set x$


Number of Parameters of Moving Average Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t$ be the deviation from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''moving average model''' on $S$ of order $q$: :$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ Then $M$ has $q + 2$ parameters. </onlyinclude>


Characteristic of Field by Annihilator/Characteristic Zero

Suppose that: :$\map {\mathrm {Ann} } F = \set 0$ That is, the annihilator of $F$ consists of the zero only. Then: :$\Char F = 0$ That is, the characteristic of $F$ is zero.


Characteristic of Field by Annihilator/Prime Characteristic

Suppose that: :$\exists n \in \map {\mathrm {Ann} } F: n \ne 0$ That is, there exists (at least one) non-zero integer in the annihilator of $F$. If this is the case, then the characteristic of $F$ is non-zero: :$\Char F = p \ne 0$ and the annihilator of $F$ consists of the set of integer multiples of $p$: :$\map {\mathrm {Ann} } F = p \Z$ where $p$ is a prime number.


Necessary Condition for Autoregressive Process to be Stationary

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''autoregessive model''' on $S$ of order $p$: :$\map \phi B \tilde z_t = a_t$ where $\map \phi B := 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ is the autoregressive operator of order $p$. Consider the polynomial equation in $B$ of degree $p$: :$(1): \quad \map \phi B = 0$ Let $\map R \phi \subseteq \C$ denote the set of roots of $(1)$, considered as a polynomial of degree $p$. It is noted that the elements of $\map R \phi$ may be real or complex. For $S$ modelled by $M$ to be a stationary process, it is necessary that the elements of $\map R \phi$ have a complex modulus greater than $1$: :$\forall z \in \map R \phi: \size z > 1$


ARIMA Model subsumes ARMA Model

Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an ARMA model for $S$. Then $M$ is also an implementation of an ARIMA model.


ARIMA Model subsumes Autoregressive Model

Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be an autoregressive model for $S$. Then $M$ is also an implementation of an ARIMA model.


Leigh.Samphier/Sandbox/Matroid Base Axiom Implies Sets Have Same Cardinality

Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. Let $\mathscr B$ satisfy the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}} Then: :$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$ where $\card{B_1}$ and $\card{B_2}$ denote the cardinality of the sets $B_1$ and $B_2$ respectively.


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5/Lemma

:$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$


Negative of Field Negative

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ and let $-a$ be the field negative of $a$. Then: :$-\paren {-a} = a$


Field Product with Zero

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$. Then: :$a \times 0_F = 0_F$


Product with Field Negative

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \in F$. Then: :$-\paren {a \times b} = a \times \paren {-b} = \paren {-a} \times b$


Product with Field Negative/Corollary

:$\paren {-1_F} \times a = \paren {-a}$


Condition for Division by Field Elements to be Unity

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b \in F$. Then: :$\dfrac a b = 1_F$ {{iff}} :$a = b$ where $\dfrac a b$ denotes division.


Field Product with Non-Zero Element yields Unique Solution

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, x \in F$ such that $b \ne 0_F$. Let: :$b \times x = a$ Then: :$x = a b^{-1}$ That is: :$x = \dfrac a b$ where $\dfrac a b$ denotes division.


Field Unity Divided by Element equals Multiplicative Inverse

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a \in F$. Then: :$\dfrac {1_F} a = a^{-1}$ where $\dfrac {1_F} a$ denotes division.


Set Difference of Larger Set with Smaller is Not Empty

Let $S$ and $T$ be finite sets. Let $\card S > \card T$. Then: :$S \setminus T \ne \O$


Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\boldsymbol \Gamma_n$ denote the '''autocovariance matrix''' associated with $S$ for $\sequence {s_n}$. Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. Then: :$\boldsymbol \Gamma_n = \sigma_z^2 \mathbf P_n$ where $\sigma_z^2$ denotes the variance of $S$.


Variance of Linear Function of Observations of Stationary Process

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $L_t$ be a linear function of $\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}$ Then the variance of $L_t$ is given by: :$\var {L_t} = \displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n l_i l_j \gamma {\size {j - i} }$ where $\gamma_k$ is the autocovariance of $S$ at lag $k$.


Autocorrelation Matrix is Positive Definite

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. Then $\mathbf P_n$ is a positive definite matrix.


Determinant of Autocorrelation Matrix is Strictly Positive

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$. The determinant of $\mathbf P_n$ is strictly positive.


Sum of Wholly Real Numbers is Wholly Real

Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers. Then $x + y$ is also wholly real.


Product of Wholly Real Numbers is Wholly Real

Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers. Then $x y$ is also wholly real.


Product of Imaginary Unit with Itself

Let $\tuple {0, 1}$ denote the imaginary unit. Then: :$\tuple {0, 1} \times \tuple {0, 1} = \tuple {-1, 0}$ where $\times$ denotes complex multiplication.


Element of Matroid Base and Circuit has a Substitute

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B \subseteq S$ be a base of $M$. Let $C \subseteq S$ be a circuit of $M$. Let $x \in B \cap C$. Then: :$\exists y \in C \setminus B : \paren{B \setminus \set x} \cup \set y$ is a base of $M$


Set Difference with Proper Subset is Proper Subset

Let $S$ be a set. Let $T \subsetneq S$ be a proper subset of $S$.   Let $S \setminus T$ denote the set difference between $S$ and $T$. Then: :$S \setminus T$ is a proper subset of $S$


Element of Matroid Base and Circuit has a Substitute/Lemma 1

:$C \setminus \set x$ is an independent proper subset of $C$


All Bases of Matroid have same Cardinality/Corollary

Let $X \subseteq S$ be any independent subset of $M$. Then: :$\card X \le \card B$


Independent Set can be Augmented by Larger Independent Set/Corollary

Let $B \subseteq S$ be a base of $M$. Then: :$\exists Z \subseteq B \setminus X : \card{X \cup Z} = \card B  : X \cup Z$ is a base of $M$


Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary

Let $B \subseteq S$ be a base of $M$. Let $X \subseteq S$ be any independent subset of $M$. Let $\card X = \card B$. Then: :$X$ is a base of $M$.


Element of Matroid Base and Circuit has a Substitute/Lemma 2

:$x \notin \paren{ C \setminus \set x} \cup X$


Element of Matroid Base and Circuit has a Substitute/Lemma 3

:$\exists y \in \paren{\paren{C \setminus \set x} \cup X} \setminus \paren{B \setminus \set x} : \paren{B \setminus \set x} \cup \set y \in \mathscr I : \card{\paren{B \setminus \set x} \cup \set y} = \card {\paren{ C \setminus \set x} \cup X}$


Matroid Unique Circuit Property

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X \subseteq S$ be an independent subset of $M$. Let $x \in S$ such that: :$X \cup \set x$ is a dependent subset of $M$. Then there exists a unique circuit $C$ such that: :$x \in C \subseteq X \cup \set x$


Absolute Value of Negative

Let $x \in \R$ be a real number. Then: :$\size x = \size {-x}$ where $\size x$ denotes the absolute value of $x$.


Dependent Subset Contains a Circuit

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr C$ denote the set of all circuits of $M$. Let $A$ be a dependent subset. Then: :$\exists C \in \mathscr C : C \subseteq A$


Condition for Linear Operation on Complex Numbers to be of Finite Order

Let $A$ be the operation on the complex numbers $\C$ defined as: :$\map A x = \alpha x + \beta$ Then $A$ is of finite order greater than $1$ {{iff}} $\alpha$ is a root of unity other than $1$.


Positive Rational Numbers under Division do not form Group

Let $\struct {\Q, /}$ denote the algebraic structure consisting of the set of rational numbers $\Q$ under the operation $/$ of division. We have that $\struct {\Q, /}$ is not a group.


Derivative of Inverse Hyperbolic Sine of x over a/Corollary 1

:$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$


Negative of Logarithm of x plus Root x squared plus a squared

Let $x \in \R$ be a real number. Then: :$-\map \ln {x + \sqrt {x^2 + a^2} } = \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}$


Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary

:$\displaystyle \int \frac {\d x} {-\sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } + C$


Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 1

:$\map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } } = \dfrac 1 {\sqrt {x^2 - a^2} }$ for $x > a$.


Derivative of Inverse Hyperbolic Cosine of x over a/Corollary 2

:$\map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } } = -\dfrac 1 {\sqrt {x^2 - a^2} }$ for $x > a$.


Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary

:$\displaystyle \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$


Derivative of Inverse Hyperbolic Tangent of x over a/Corollary

:$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$ where $\size x < a$.


Derivative of Inverse Hyperbolic Cotangent of x over a/Corollary

:$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } } = \dfrac 1 {a^2 - x^2}$ where $\size x > a$.


Leigh.Samphier/Sandbox/Set Difference of Matroid Circuit with Element is Independent

:$C \setminus \set x$ is an independent subset of $C$


Union of Matroid Base with Element of Complement is Dependent

:$B \cup \set x$ is a dependent superset of $B$


Primitive of Power of a x + b/Proof 2

{{:Primitive of Power of a x + b}} where $n \ne 1$.


Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form

:$\displaystyle \int \dfrac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\ & \\ \dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\ & \\ \text {undefined} & : x = a \end {cases}$


Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form

:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \frac 1 a \coth^{-1} \frac x a + C$ where $\size x > a$.


Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a

Let $\size x < a$. Then: :$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$


Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a

Let $\size x > a$. Then: :$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C$


Primitive of Reciprocal of a squared minus x squared/Logarithm Form

Let $a \in \R_{>0}$ be a strictly positive real constant. Let $x \in \R$ such that $\size x \ne a$.


Primitive of Reciprocal of x squared minus a squared/Logarithm Form


Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1

:$\displaystyle \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\ & \\ \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$


Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a

Let $\size x < a$. Then: :$\displaystyle \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$


Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x greater than a

Let $\size x > a$. Then: :$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C$


Power Set is Closed under Complement

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A \in \powerset S: \relcomp S A \in \powerset S$ where $\relcomp S A$ denotes the complement of $A$ relative to $S$.


Parallelism is Reflexive Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a reflexive relation on $S$.


Parallelism is Symmetric Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a symmetric relation on $S$.


Parallelism is Equivalence Relation/Transitivity

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is a transitive relation on $S$.


Approximation/Examples/22 over 7

$\dfrac {22} 7$ is a convenient approximation to $\pi$: :$\dfrac {22} 7 = 3 \cdotp \dot 14285 \dot 7$


Perpendicularity is Antireflexive Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is an antireflexive relation on $S$.


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma

:$\exists B_3 \in \mathscr B$: :::$V \subseteq B_3$ :::$\card{B_1 \cap B_3} > \card{B_1 \cap B_2}$


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 2

{{begin-eqn}} {{eqn | l = \card {B_1}       | r = \card{B_1 \cap B_2} + \card{B_1 \setminus B_2} }} {{eqn | l = \card {B_2}       | r = \card{B_2 \cap B_1} + \card{V \setminus B_1} + \card{\paren{B_2 \setminus B_1} \setminus V} }} {{end-eqn}}


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 3

:$\card{B_1 \cap B_3} = \card{B_1 \cap B_2} + 1$


Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1

The following formulation of the Axiom of Choice:


Equivalence of Versions of Axiom of Choice/Formulation 1 implies Formulation 3


Equivalence of Versions of Axiom of Choice/Formulation 3 implies Formulation 1


C^k Norm is Norm

Let $I = \closedint a b$ be a closed real interval. Let $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, k-times differentiable on $I$. Let $x \in \map {\CC^k} I$ be a real-valued function of differentiability class $k$. Let $\norm {\, \cdot \,}_{\map {C^k} I}$ be the $C^k$ norm on $I$. Then $\norm {\, \cdot \,}_{\map {C^k} I}$ is a norm on $\struct {\map {\CC^k} I, +, \, \cdot \,}_\R$.


Existence of Minimal Polynomial for Square Matrix over Field

Let $K$ be a field. Let $n$ be a natural number. Let $K^{n \times n}$ be the set of $n \times n$ matrices over $K$.  Let $A \in K^{n \times n}$. Then the minimal polynomial of $A$ exists and has degree at most $n^2$.


Homeomorphic Topologies on Same Set may not be Identical

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ both be topologies on $S$ such that the topological spaces $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic. Then it is not necessarily the case that $\struct {S, \tau_1} = \struct {S, \tau_1}$.


Clopen Sets in Indiscrete Topology

The only subsets of $S$ which are both closed and open in $T$ are $S$ and $\O$.


Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets

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


Dependent Subset of Independent Set Union Singleton Contains Singleton

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X$ be an independent subset of $M$. Let $x \in S$. Let $C$ be a dependent subset of $M$ such that: :$C \subseteq X \cup \set x$. Then: :$x \in C$


Matroid Unique Circuit Property/Corollary

Let $B$ be a base of $M$. Let $x \in S \setminus B$. Then there exists a unique circuit $C$ such that: :$x \in C \subseteq B \cup \set x$ That is, $C$ is the fundamental circuit of $x$ in $B$.


Leigh.Samphier/Sandbox/Matroid Base Substitution From Fundamental Circuit

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $B$ be a base of $M$. Let $y \in S \setminus B$. Let $\map C {y,B}$ denote the fundamental circuit of $y$ in $B$. Let $x \in B$. Then: :$\paren{B \setminus \set x} \cup \set y$ is a base of $M$ {{iff}} $x \in \map C {y,B}$


Socrates is Mortal/Variant

:$(1): \quad$ ''If {{AuthorRef|Socrates}} is a man then {{AuthorRef|Socrates}} is mortal.'' :$(2): \quad$ ''{{AuthorRef|Socrates}} is a man.'' :$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.''


Self-Distributive Law for Conditional/Forward Implication/Formulation 2

:$\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$


Self-Distributive Law for Conditional/Reverse Implication/Formulation 2

:$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$


Odd Integers under Addition do not form Group

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


Number of Digits in Number

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $b \in \Z_{>1}$ be an integer greater than $1$. Let $n$ be expressed in base $b$. Then the number of digits $d$ in this expression for $n$ is: :$d = 1 + \floor {\log_b n}$ where: :$\floor {\, \cdot \,}$ denotes the floor function :$\log_b$ denotes the logarithm to base $b$.


Every Tenth Power of Two Minus Every Third Power of Ten is Divisible By Three

Let $x \in \Z_{\ge 0}$ be a non-negative integer. Then $2^{10 x} - 10^{3 x}$ is divisible by $3$. That is: :$2^{10 x} - 10^{3 x} \equiv 0 \pmod 3$


Gödel's Incompleteness Theorems/Second

Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic. Let $\map {\mathrm {Cons} } T$ be the propositional function which states that $T$ is consistent. Then it is not possible to prove $\map {\mathrm {Cons} } T$ by means of formal statements within $T$ itself.


Linear Function on Stationary Stochastic Model is Stationary

Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$. Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$: :$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$ Let $L_t$ be a linear function of $\sequence {s_n}$: :$L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}$ Then $L_t$ is itself stationary.


Strict Ordering on Integers is Well-Defined

Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Let: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | r = \eqclass {a', b'} {}       | c =  }} {{eqn | l = \eqclass {c, d} {}       | r = \eqclass {c', d'} {}       | c =  }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | ll= \iff       | l = \eqclass {a', b'} {}       | o = <       | r = \eqclass {c', d'} {}       | c =  }} {{end-eqn}}


Definition:Gaussian Process

Let $S$ be a stochastic process giving rise to a time series $T$. Let the probability distribution of $T$ be a multivariate normal distribution. Then $S$ is called a '''Gaussian process'''.


Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary

Let $S$ be a Gaussian stochastic process giving rise to a time series $T$. Let $S$ be weakly stationary of order $2$. Then $S$ is strictly stationary of order $2$.


Ordering on Integers is Transitive

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


Strict Ordering on Integers is Asymmetric

Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers. Then: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | lo= \implies       | l = \eqclass {c, d} {}       | o = \not <       | r = \eqclass {a, b} {}       | c =  }} {{end-eqn}} That is, strict ordering on the integers is asymmetric.


Negative of Integer

Let $x \in \Z$ be an integer. Let $x = \eqclass {a, b} {}$ be defined from the formal definition of integers, where $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers. Then: :$-x = \eqclass {b, a} {}$


Symmetric Function Theorem

Let $f$ be a polynomial in $n$ variables. Let $f$ be of degree $r$ in each of its $n$ variables. Then $f$ is equal to a polynomial of total degree $r$ with integer coefficients in the elementary symmetric functions: :$ds \sum x_i \sim x_i x_j, \dotsc, \prod x_j$ and the coefficients of $f$.


Euler's Integral Theorem

:$\ds H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$ where: :$H_n$ denotes the $n$th harmonic number :$\gamma$ denotes the Euler-Mascheroni constant.


Ordering on Positive Integers is Equivalent to Ordering on Natural Numbers

Let $u, v \in \Z_{>0}$ be natural numbers. Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as: :$\forall u \in \N_{>0}: \map \phi u = u'$ where $u' \in \Z$ denotes the (strictly) positive integer $\eqclass {b + u, b} {}$. Let $u', v' \in \Z_{>0}$ be strictly positive integers. Then: :$u > v \iff u' > v'$


Product of Absolute Values of Integers

Let $a, b \in \Z$ be integers. Let $\size a$ denote the absolute value of $a$: :$\size a = \begin {cases} a & : a \ge 0 \\ -a : a < 0 \end {cases}$ Then: :$\size a \times \size b = \size {a \times b}$


Displacement of Particle under Force

Let $P$ be a particle of constant mass $m$. Let the position of $P$ at time $t$ be specified by the position vector $\mathbf r$. Let a force applied to $P$ be represented by the vector $\mathbf F$. Then the motion of $P$ can be given by the differential equation: :$\mathbf F = m \dfrac {\d^2 \mathbf r} {\d t^2}$ or using Newtonian notation: :$\mathbf F = m \ddot {\mathbf r}$


1-Sequence Space is Separable

$\ell^1$ space is a separable space.


Primitive of Root of Function under Half its Derivative

Let $f$ be a real function which is integrable. Then: :$\ds \int \frac {\map {f'} x} {2 \sqrt {\map f x} } \rd x = \sqrt {\map f x} + C$ where $C$ is an arbitrary constant.


Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form

:$\ds \int \sech x \rd x =  2 \map \arctan {\tanh \dfrac x 2} + C$


First Order ODE/y' + 2 x y = 1

The first order ODE: :$y' + 2 x y = 1$ has the general solution: :$y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$ where $a$ is an arbitrary constant.


Electric Potential over Conducting Surface is Constant

Let $S$ be a conducting surface. The electric potential $V$ over $S$ is constant. This can be expressed using the Laplacian: :$\nabla^2 V = 0$ and is thus seen to satisfy Laplace's equation.


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 4

Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 3 Iff Definition 7


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3


P-Norm is Norm/P-Sequence Space

The $p$-norm on the $p$-sequence space is a norm.


P-Norm is Norm/Complex Numbers

The $p$-norm on the complex numbers is a norm.


P-Norm is Norm/Real Numbers

The $p$-norm on the real numbers is a norm.


Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space

Let $\ell^\infty$ be the space of bounded sequences. Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers. Let $\paren +$ be the pointwise addition on the ring of sequences. Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences. Then $\struct {\ell^\infty, +, \, \cdot \,}_\C$ is a vector space.


Zero Vector has no Direction

A zero vector has no direction.


Power Set is Closed under Countable Unions

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A_n \in \powerset S: n = 1, 2, \ldots: \ds \bigcup_{n \mathop = 1}^\infty A_n \in \powerset S$


Probability of Union of Disjoint Events is Sum of Individual Probabilities

Let $\EE$ be an experiment. Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\EE$. Then: :$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$


Probability Measure on Finite Sample Space

Let $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ be a finite set. Let $\Sigma$ be a $\sigma$-algebra on $\Omega$. Let $p_1, p_2, \ldots, p_n$ be non-negative real numbers such that: :$p_1 + p_2 + \cdots + p_n = 1$ Let $Q: \Sigma \to \R$ be the mapping defined as: :$\forall A \in \Sigma: \map Q A = \ds \sum_{i: \omega_i \in A} p_i$ Then $\struct {\Omega, \Sigma, Q}$ constitutes a probability space. That is, $Q$ is a probability measure on $\struct {\Omega, \Sigma}$.


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms

Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers. {{TFAE|def=Rank Axioms (Matroid)|view=Matroid Rank Axioms}}


Length of Element of Arc in Orthogonal Curvilinear Coordinates

Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates. Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where $\tuple {x, y, z}$ denotes the Cartesian coordinates. The length $\d l$ of a small arc is given by: :$\d l = {h_1}^2 {\d q_1}^2 + {h_2}^2 {\d q_2}^2 + {h_3}^2 {\d q_3}^2$ where: :${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$


Laplacian of Function in Orthogonal Curvilinear Coordinates

Let $\map \psi {q_1, q_2, q_3}$ denote a real-valued function embedded in an orthogonal curvilinear coordinate system. Then the Laplacian of $\psi$ can be expressed as: :$\nabla^2 \psi = \dfrac 1 {h_1 h_2 h_3} \paren {\map {\dfrac \partial {\partial q_1} } {\dfrac {h_2 h_3} {h_1} \dfrac {\partial \psi} {\partial q_1} } + \map {\dfrac \partial {\partial q_2} } {\dfrac {h_3 h_1} {h_2} \dfrac {\partial \psi} {\partial q_2} } + \map {\dfrac \partial {\partial q_3} } {\dfrac {h_1 h_2} {h_3} \dfrac {\partial \psi} {\partial q_3} } }$ where: :${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$


Pressure of Gas for Density and Temperature

The pressure, density and temperature of a body of gas are linked by the equation: :$p = k \rho \paren {1 + \alpha t}$ where: :$p$ denotes the pressure of the gas :$\rho$ denotes the density of the gas :$t$ denotes thetemperature of the gas :$k$ and $\alpha$ are constants.


Leigh.Samphier/Sandbox/Matroid Satisfies Circuit Axioms

Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. Then $\mathscr C$ is the set of circuits of a matroid on $S$ {{iff}} $\mathscr C$ satisfies the circuit axioms: {{: Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}}


Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms

Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. {{TFAE|def=Circuit Axioms (Matroid)|view=Matroid Circuit Axioms}}


Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Let $\rho: \powerset S \to \Z$ denote the rank function of $M$. Then: :$\map \rho C = \card C -1$


Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Let $\rho: \powerset S \to \Z$ denote the rank function of $M$. Then: :$\card C \le \map \rho S + 1$


Leigh.Samphier/Sandbox/Matroid with No Circuits Has Single Base

Let $M = \struct {S, \mathscr I}$ be a matroid with no circuits. Then: :$S$ is the only base on $M$.


Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Then: :every proper subset $A$ of $C$ is independent


Supremum Norm is Norm/Space of Bounded Sequences

The supremum norm on the space of bounded sequences is a norm.


Parallelogram Law for Vector Subtraction

Let $\mathbf u$ and $\mathbf v$ be vectors. Consider a parallelogram, two of whose adjacent sides represent $\mathbf y$ and $\mathbf v$ (in magnitude and direction). :400px Then the diagonal of the parallelogram connecting the terminal points of $\mathbf u$ and $\mathbf v$ represents the magnitude and direction of $\mathbf u - \mathbf v$, the difference of $\mathbf u$ and $\mathbf v$.


Space of Bounded Sequences with Supremum Norm forms Normed Vector Space

Space of bounded sequences with supremum norm forms a normed vector space.


Riesz's Lemma

Let $X$ be a normed vector space. Let $Y$ be a proper closed linear subspace of $X$.  Let $\alpha \in \openint 0 1$. Then there exists $x_\alpha \in X$ such that:  :$\norm {x_\alpha} = 1$ with: :$\norm {x_\alpha - y} > \alpha$ for all $y \in Y$.


Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact

Let $X$ be a normed vector space. Let $S = \map {S_1} 0$ be the unit sphere centred at $0$ in $X$. Then $X$ is finite dimensional {{iff}} $S$ is compact.


Finite Dimensional Subspace of Normed Vector Space is Closed

Let $V$ be a normed vector space. Let $W$ be a finite dimensional subspace of $V$.  Then $W$ is closed.


Catalan's Conjecture

The only solution to the Diophantine equation: :$x^a - y^b = 1$ for $a, b > 1$ and $x, y > 0$, is: :$x = 3, a = 2, y = 2, b = 3$


Total Force on Charged Particle from 2 Charged Particles

Let $p_1$, $p_2$ and $p_3$ be charged particles. Let $q_1$, $q_2$ and $q_3$ be the electric charges on $p_1$, $p_2$ and $p_3$ respectively. Let $\mathbf F_{ij}$ denote the force exerted on $q_j$ by $q_i$. Let $\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of the other two charged particles. Then the force $\mathbf F_1$ exerted on $q_1$ by the combined action of $q_2$ and $q_3$ is given by: {{begin-eqn}} {{eqn | l = \mathbf F_1       | r = \mathbf F_{21} + \mathbf F_{31}       | c =  }} {{eqn | r = \dfrac {q_2 q_1} {4 \pi \varepsilon_0 r_{2 1}^3} \mathbf r_{2 1} + \dfrac {q_3 q_1} {4 \pi \varepsilon_0 r_{3 1}^3} \mathbf r_{3 1}       | c =  }} {{end-eqn}} where: :$\mathbf F_{21} + \mathbf F_{31}$ denotes the vector sum of $\mathbf F_{21}$ and $\mathbf F_{31}$ :$\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\varepsilon_0$ denotes the vacuum permittivity.


Total Force on Charged Particle from Multiple Charged Particles

Let $p_1, p_2, \ldots, p_n$ be charged particles. Let $q_1, q_2, \ldots, q_n$ be the electric charges on $p_1, p_2, \ldots, p_n$ respectively. For all $i$ in $\set {1, 2, \ldots, n}$ where $i \ne j$, let $\mathbf F_{i j}$ denote the force exerted on $q_j$ by $q_i$. For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of all the other charged particles. Then the force $\mathbf F_i$ exerted on $q_i$ by the combined action of all the other charged particles is given by: {{begin-eqn}} {{eqn | l = \mathbf F_i       | r = \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \mathbf F_{j i}       | c =  }} {{eqn | r = \dfrac 1 {4 \pi \varepsilon_0} \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \dfrac {q_i q_j} {r_{j i}^3} \mathbf r_{j i}       | c =  }} {{end-eqn}} where: :the summation denotes the vector sum of $\mathbf F_{21}$ and $\mathbf F_{31}$ :$\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\varepsilon_0$ denotes the vacuum permittivity.


Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space

Space of Continuously Differentiable on Closed Interval Real-Valued Functions with $C^1$ norm forms a normed vector space.


Ring of Endomorphisms is Ring with Unity

Let $\struct {G, \oplus}$ be an abelian group. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $\struct {\mathbb G, \oplus, *}$ denote the '''ring of endomorphisms''' on $\struct {G, \oplus}$. Then $\struct {\mathbb G, \oplus, *}$ is a ring with unity.


Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm is Banach Space

Let $I := \closedint a b$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on $I$. Let $\map {\CC^1} I$ be the space of real-valued functions, continuously differentiable on $I$. Let $\norm {\, \cdot \,}_{1, \infty}$ be the $\CC^1$ norm. $\struct {\map {\CC^1} I, \norm {\, \cdot \,}_{1, \infty} }$ be the normed space of real-valued functions, continuously differentiable on $I$. Then $\struct {\map {\CC^1} I, \norm {\, \cdot \,}_{1, \infty} }$ is a Banach space.


Set of Order m times n Matrices does not form Ring

Let $m, n \in \N_{>0}$ be non-zero natural numbers such that $m > n$. Let $S$ be the set of all matrices of order $m \times n$. Then the algebraic structure $\struct {S, +, \times}$ is not a ring. Note that $\times$ denotes conventional matrix multiplication.


Set of Order 3 Vectors under Cross Product does not form Ring

Let $S$ be the set of all vectors in a vector space of dimension $3$. Let $\times$ denote the cross product operation. Then the algebraic structure $\struct {S, +, \times}$ is not a ring.


Integers under Subtraction do not form Semigroup

Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\struct {\Z, -}$ is not a semigroup.


Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Necessary Condition

Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be the rank function of a matroid on $S$. Then $\rho$ satisfies the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}}


Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition

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


Leigh.Samphier/Sandbox/Independent Subset is Contained in Maximal Independent Subset/Corollary

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $X$ be a maximal independent subset of $A$. Then: :$\card X = \map \rho A$ where $\rho$ is the rank function on $M$.


Motion of Body with Constant Mass

Let $B$ be a body with constant mass $m$ undergoing a force $\mathbf F$. Then the equation of motion of $B$ is given by: :$\mathbf F = m \mathbf a$ where $\mathbf a$ is the acceleration of $B$.


Like Vector Quantities are Multiples of Each Other

Let $\mathbf a$ and $\mathbf b$ be like vector quantities. Then: :$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ denotes the scalar product of $\mathbf b$ by $\dfrac {\size {\mathbf a} } {\size {\mathbf b} }$.


Direct Product Norm is Norm

Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\norm {\tuple {x, y} }$ be the direct product norm. Then $\norm {\tuple {x, y} }$ is a norm on $V$.


Vector Quantity as Scalar Product of Unit Vector Quantity

Let $\mathbf a$ be a vector quantity. Then: :$\mathbf a = \size {\mathbf a} \mathbf {\hat a}$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\mathbf {\hat a}$ denotes the unit vector in the direction $\mathbf a$.


Scalar Product of Magnitude by Unit Vector Quantity

Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. Then: :$m \mathbf a = m \paren {\size {\mathbf a} \hat {\mathbf a} } = \paren {m \size {\mathbf a} } \hat {\mathbf a}$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\hat {\mathbf a}$ denotes the unit vector in the direction $\mathbf a$.


Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors

Let $\mathbf r$ be a vector quantity embedded in space. Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be non-coplanar. Then $\mathbf r$ can be expressed uniquely as the resultant of $3$ vector quantities which are each parallel to one of $\mathbf a$, $\mathbf b$ and $\mathbf c$.


Vectors are Equal iff Components are Equal

Two vector quantities are equal {{iff}} they have the same components.


Components of Vector in terms of Direction Cosines

Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $x$, $y$ and $z$ be the components of $\mathbf r$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions respectively. Let $r$ denote the magnitude of $\mathbf r$, that is: :$r := \size {\mathbf r}$ Then: {{begin-eqn}} {{eqn | l = x       | r = r \cos \alpha }} {{eqn | l = y       | r = r \cos \beta }} {{eqn | l = z       | r = r \cos \gamma }} {{end-eqn}}


Magnitude of Vector Quantity in terms of Components

Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf r$ be expressed in terms of its components: :$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$ where $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\size {\mathbf r} = \sqrt {x^2 + y^2 + z^2}$ where $\size {\mathbf r}$ denotes the magnitude of $\mathbf r$.


Components of Zero Vector Quantity are Zero

Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf r$ be expressed in terms of its components: :$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$ Let $\mathbf r$ be the zero vector. Then: :$x = y = z = 0$


Unit Vector in terms of Direction Cosines

Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $\mathbf {\hat r}$ denote the unit vector in the direction of $\mathbf r$. Then: :$\mathbf {\hat r} = \paren {\cos \alpha} \mathbf i + \paren {\cos \beta} \mathbf j + \paren {\cos \gamma} \mathbf k$


Unit Vectors of Cartesian 3-Space form Basis

Consider the Cartesian $3$-space $C$. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then $\set {\mathbf i, \mathbf j, \mathbf k}$ forms a basis for $C$.


Vectors from Sum and Difference

Let $\mathbf a$ and $\mathbf b$ be vector quantities. Let $\mathbf c = \mathbf a + \mathbf b$ and $\mathbf d = \mathbf a - \mathbf b$ be given. Then: {{begin-eqn}} {{eqn | l = \mathbf a       | r = \dfrac 1 2 \paren {\mathbf c + \mathbf d} }} {{eqn | l = \mathbf b       | r = \dfrac 1 2 \paren {\mathbf c - \mathbf d} }} {{end-eqn}}


Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition/Lemma

:$\forall Y \subseteq S: \map \rho Y \le \card Y$


Dot Product of Perpendicular Vectors

Let $\mathbf a$ and $\mathbf b$ be vector quantities. Let $\mathbf a$ and $\mathbf b$ be perpendicular. Then: :$\mathbf a \cdot \mathbf b = 0$ where $\cdot$ denotes dot product.


Dot Product of Orthogonal Basis Vectors

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be an orthonormal basis of a vector space $V$. Then: :$\forall i, j \in \set {1, 2, \ldots, n}: \mathbf e_i \cdot \mathbf e_j = \delta_{i j}$ where: :$\mathbf e_i \cdot \mathbf e_j$ denotes the dot product of $\mathbf e_i$ and $\mathbf e_j$ :$\delta_{i j}$ denotes the Kronecker delta.


Second Partial Derivative wrt r of ln (r^2 + s)

:$\dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s} = \dfrac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$


Derivative of Square of Tangent

:$\map {\dfrac \d {\d x} } {\tan^2 x} = 2 \tan x \sec^2 x$ when $\cos x \ne 0$.


Partial Derivatives of x^y^z

Let: :$u = x^{\paren {y^z} }$ Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial x}       | r = y^z x^{\paren {y^z - 1} } }} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = x^{y^z} z y^{z - 1} \ln x }} {{eqn | l = \dfrac {\partial u} {\partial z}       | r = x^{\paren {y^z} } y^z \ln x \ln y }} {{end-eqn}}


Partial Derivatives of x^u + u^y

Let: :$u = x^u + u^y$ Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial x}       | r = \frac {u^2} {x \paren {1 - u \ln x - y} } }} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = \frac {u \ln u} {1 - u \ln x - y} }} {{end-eqn}}


Third Partial Derivatives of x^y

Let: :$u = x^y$ Then: :$\dfrac {\partial^3 u} {\partial x^2 \partial y} = \dfrac {\partial^3 u} {\partial x \partial y \partial x}$


Number of Terms in Homogeneous Polynomial

The number of terms in a homogeneous polynomial of degree $n$ in $m$ indeterminates is given by: :$N = \dbinom {n + m - 1} n = \dfrac {\paren {n + m - 1}!}{n! \, \paren {m - 1}!}$


Definition:Set

A '''set''' is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other. We can think of each set as a single entity in itself, and we can denote it (and usually do) by means of a single symbol. That is, ''anything you care to think of'' can be a set. This concept is known as the comprehension principle. However, there are problems with the comprehension principle. If we allow it to be used without any restrictions at all, paradoxes arise, the most famous example probably being Russell's Paradox. Hence some sources define a '''set''' as a ''' 'well-defined' collection of objects''', leaving the concept of what constitutes well-definition to later in the exposition.


Definition:Element

Let $S$ be a set. An '''element of $S$''' is a member of $S$.


Definition:Zermelo-Fraenkel Axioms

=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} === The Axiom of Replacement === {{:Axiom:Axiom of Replacement}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation}}


Definition:Empty Set

The '''empty set''' is a set which has no elements. That is, $x \in \O$ is false, whatever $x$ is.


Definition:Singleton

A '''singleton''' is a set that contains exactly one element.


Definition:Subset

Let $S$ and $T$ be sets. $S$ is a '''subset''' of a set $T$ {{iff}} all of the elements of $S$ are also elements of $T$. This is denoted: :$S \subseteq T$ That is: :$S \subseteq T \iff \forall x: \paren {x \in S \implies x \in T}$ If the elements of $S$ are not all also elements of $T$, then $S$ is not a '''subset''' of $T$: :$S \nsubseteq T$ means $\neg \paren {S \subseteq T}$


Definition:Set Union

Let $S$ and $T$ be sets. The '''(set) union''' of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$: :$x \in S \cup T \iff x \in S \lor x \in T$


Definition:Set Intersection

Let $S$ and $T$ be sets. The '''(set) intersection''' of $S$ and $T$ is written $S \cap T$. It means the set which consists of all the elements which are contained in both of $S$ and $T$: :$x \in S \cap T \iff x \in S \land x \in T$


Definition:Set Difference

The '''(set) difference''' between two sets $S$ and $T$ is written $S \setminus T$, and means the set that consists of the elements of $S$ which are not elements of $T$: :$x \in S \setminus T \iff x \in S \land x \notin T$


Definition:Set Equality

Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Set Equality/Definition 1}} === Definition 2 === {{:Definition:Set Equality/Definition 2}}


Definition:Statement

A '''statement''' is a sentence which has objective and logical meaning.


Definition:Proposition

A '''proposition''' is a statement which is offered up for investigation as to its truth or falsehood. Loosely, a '''proposition''' is a statement which is about to be proved (or disproved).


Definition:Truth Value

=== Aristotelian Logic === {{:Definition:Truth Value/Aristotelian Logic}} {{expand|Other logic types}}


Definition:Logic

'''Logic''' is the study of the structure of statements and their truth values, divorced from their conceptual content. It has frequently been defined as ''' the science of reasoning'''.


Definition:Atom (Logic)

In a particular branch of logic, certain concepts are at such a basic level of simplicity they can not be broken down into anything simpler. Those concepts are called '''atoms''' or described as '''atomic'''. Different branches of logic admit different '''atoms'''. === Propositional Logic === In propositional logic, the '''atoms''' are statements.


Definition:Subject

The '''subject''' of a simple statement in logic is the part of the statement specifying the object being talked about. The '''subject''' of a simple statement is atomic in predicate logic. The '''subject''' and predicate of a simple statement are referred to as its terms.


Definition:Predicate

The '''predicate''' of a simple statement in logic is the part of the statement which defines ''what is being said'' about the subject. It is a word or phrase which, when combined with one or more names of objects, turns into a meaningful sentence. The predicates of simple statements are atomic in predicate logic. The subject and predicate of a simple statement are referred to as its terms.


Definition:Simple Statement

A '''simple statement''' is a statement which has one subject and one predicate. For example, the statement: :'''London is the capital of England''' is a '''simple statement'''. '''London''' is the subject and '''is the capital of England''' is the predicate.


Definition:Logical Term

Both the subject and the predicate of a simple statement are referred to as its '''(logical) terms'''. Note that this use of the word '''term''' is a more specialized use of the word term as used in algebra.


Definition:Object

An '''object''' is a ''thing''. Everything that can be talked about in mathematics and logic can be referred to as an '''object'''.


Definition:Symbol

In its broadest possible sense: :A '''symbol''' is an object used to represent another object. In a narrower and more "mathematical" sense, a '''symbol''' is a sign of a particular shape to which is assigned a meaning, and is used to represent a concept or identify a particular object. It is generally much more convenient to use a symbol than the plain speaking that it replaces, because it is invariably more compact. One character can replace a large number of words. As definitions become more complex, the symbols tend to convey more information -- but by the same coin, understanding exactly what a symbol means becomes more difficult. Symbols may mean different things in different contexts. A symbol that means something in one context may mean something completely different in another. This is because the number of different concepts is greater than human ingenuity can create symbols for, and some of them naturally have to be used more than once. This does not matter as long as, before we use any symbol, we define exactly what we mean by it. Some symbols are standard and rarely need defining, but in some contexts there are subtle differences to the ''exact'' meaning of a "standard" symbol. Therefore all fields of mathematics generally introduce themselves with a rash of definitions, many of which are symbols.


Definition:Variable/Predicate Logic

In the context of predicate logic, a variable is often called an '''object variable''' or '''arbitrary name'''. As such, it is a symbol which is assigned to an ''arbitrarily selected'' object from a given universe of discourse. The understanding is that (during the scope of the argument to which it is relevant) the arbitrary name could apply equally well to ''any'' of the objects in that universe.


Definition:Operation/Arity

The '''arity''' of an operation is the number of operands it uses. The '''arity''' of an operation may be, in general, any number. It may even be infinite.


Definition:Function

The process which is symbolised by an operation is called a '''function'''. The operand(s) of the operation can be considered to be the input(s). The output of the '''function''' is whatever the operation is defined as doing with the operand(s). A '''function''' is in fact another name for a mapping, but while the latter term is used in the general context of set theory and abstract algebra, the term '''function''' is generally reserved for mappings between sets of numbers.


Definition:Equals

:$x = y$ means '''$x$ is the same object as $y$''', and is read '''$x$ equals $y$''', or '''$x$ is equal to $y$'''. :$x \ne y$ means '''$x$ is not the same object as $y$''', and is read '''$x$ is not equal to $y$'''. The expression: :$a = b$ means: :$a$ and $b$ are names for the same object.


Definition:Symbolic Logic

'''Symbolic logic''' is the study of logic in which the logical form of statements is analyzed by using symbols as tools. Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular. In '''symbolic logic''', the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field.


Definition:Propositional Logic

'''Propositional logic''' is a sub-branch of symbolic logic in which the truth values of propositional formulas are investigated and analysed. The atoms of propositional logic are simple statements. There are various systems of propositional logic for determining the truth values of propositional formulas, for example: * Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents. * The Method of Truth Tables, which consists of the construction of one or more truth tables which exhaustively list all the possible truth values of all the statement variables with a view to determining the required answer by inspection.


Definition:Predicate Logic

'''Predicate logic''' is a sub-branch of symbolic logic. It is an extension of propositional logic in which the internal structure of simple statements is analyzed. Thus in '''predicate logic''', simple statements are no longer atomic. The atoms of '''predicate logic''' are subjects and predicates of simple statements. There are various formal systems allowing for rigid determination of the theorems of '''predicate logic''':


Definition:Statement Label

A '''statement label''' is a symbol which is assigned to a ''particular'' statement, so that it can be identified without the need to write it out in full. For such conventions to make sense, different statements must always be given different '''statement labels'''. The citing of a statement label can be interpreted as an assertion that the statement represented by that symbol is true. That is: :'''$P$''' means :'''$P \text { is true}$'''


Definition:Logical Connective

A '''logical connective''' is an object which either modifies a statement, or combines existing statements into a new statement, called a compound statement. It is almost universal to identify a '''logical connective''' with the symbol representing it. Thus, '''logical connective''' may also, particularly in symbolic logic, be used to refer to that symbol, rather than speaking of a '''connective symbol''' separately. In mathematics, '''logical connectives''' are considered to be '''truth-functional'''.  That is, the truth value of a compound statement formed using the '''connective''' is assumed to depend ''only'' on the truth value of the comprising statements. Thus, as far as the '''connective''' is concerned, it does not matter what the comprising statements precisely ''are''. As a consequence of this truth-functionality, a '''connective''' has a corresponding truth function, which goes by the same name as the '''connective''' itself. The arity of this truth function is the number of statements the '''logical connective''' combines into a single compound statement. === Unary Logical Connective === {{:Definition:Logical Connective/Unary}} === Binary Logical Connective === {{:Definition:Logical Connective/Binary}}


Definition:Natural Language

A '''natural language''' is one of the conventional, everyday languages in which people usually communicate. Although there are many natural languages in the world, we are not generally going to distinguish between them, merely lumping them all into the one concept.  When '''natural language''' is referred to on {{ProofWiki}}, it will usually mean English.


Definition:Compound Statement

A '''compound statement''' is a statement which results from the application of one or more logical connectives to a collection of simple statements. === Substatement === {{:Definition:Compound Statement/Substatement}} === Ill-Formed Compound Statement === {{:Definition:Compound Statement/Ill-Formed}}


Definition:Compound Statement/Substatement

A '''substatement''' of a compound statement is one of the statements that comprise it.


Definition:Statement Form

A '''statement form''' is a symbolic representation of a compound statement. It consists of statement variables along with logical connectives joining them. It is traditional, particularly in the field of mathematical logic, to use lowercase Greek letters to stand for general formulas (the usual ones being $\phi, \psi$ and $\chi$), but more modern treatments are starting to use ordinary lowercase letters of the English alphabet, usually $p, q, r$ etc. === Specific Form === {{:Definition:Statement Form/Specific Form}}


Definition:Logical Formula

Let $\mathcal L$ be a formal language used in the field of symbolic logic. Then the well-formed formulas of $\mathcal L$ are often referred to as '''logical formulas'''. They are symbolic representations of statements, and often of compound statements in particular.


Definition:Logical Connective/Unary

A '''unary logical connective''' (or '''one-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''one'' substatement. In standard Aristotelian logic, there are four of these. The only non-trivial one is logical not, as shown on Unary Truth Functions.


Definition:Logical Connective/Binary

A '''binary logical connective''' (or '''two-place connective''') is a connective whose effect on its compound statement is determined by the truth value of ''two'' substatements. In standard Aristotelian logic, there are 16 '''binary logical connectives''', cf. Binary Truth Functions. In the field of symbolic logic, the following four (symbols for) '''binary logical connectives''' are commonly used: * Conjunction: the '''And''' connective $p \land q$: '''$p$ is true ''and'' $q$ is true'''. * Disjunction: the '''Or''' connective $p \lor q$: '''$p$ is true ''or'' $q$ is true, ''or possibly both'''''. * The conditional connective $p \implies q$: '''''If'' $p$ is true, ''then'' $q$ is true'''. * The biconditional connective $p \iff q$: '''$p$ is true ''if and only if'' $q$ is true''', or '''$p$ ''is equivalent to'' $q$'''.


Definition:Axiom

In all contexts, the definition of the term '''axiom''' is by and large the same. That is, an '''axiom''' is a statement which is ''accepted'' as being true. A statement that is considered an '''axiom''' can be described as being '''axiomatic'''.


Definition:Logical Argument

A '''logical argument''' (or just '''argument''') is a process of creating a new statement from one or more existing statements. An '''argument''' proceeds from a set of premises to a conclusion, by means of logical implication, via a procedure called logical inference. An '''argument''' may have more than one premise, but only one conclusion. While statements may be classified as either '''true''' or '''false''', an '''argument''' may be classified as either valid or invalid.


Definition:Assumption

An '''assumption''' is a statement which is introduced into an argument, whose truth value is (temporarily) accepted as True. In mathematics, the keyword '''let''' is often the indicator here that an assumption is going to be introduced. For example: :'''Let $p$''' (be true) ... can be interpreted, in natural language, as: :'''Let us assume, for the sake of argument, that $p$ is true''' ...


Definition:Premise

A '''premise''' is an assumption that is used as a basis from which to start to construct an argument. When the validity or otherwise of a proof is called into question, one may request the arguer to "check your premises".


Definition:Conclusion

A '''conclusion''' is a statement that is obtained as the result of the process of an argument.


Definition:Valid Argument

A '''valid argument''' is a logical argument in which the premises provide conclusive reasons for the conclusion. When a proof is valid, we may say one of the following: * The conclusion '''follows from''' the premises; * The premises '''entail''' the conclusion; * The conclusion is true '''on the strength of''' the premises; * The conclusion is '''drawn from''' the premises; * The conclusion is '''deduced from''' the premises; * The conclusion is '''derived from''' the premises.


Definition:Proof

A '''proof''' is another name for a valid argument, but in this context the assumption is made that the premises are all true. That is, a valid argument that has one or more false premises is not a proof.


Definition:Invalid Argument

An '''invalid argument''' is a argument in which the premises do ''not'' provide conclusive reasons for the conclusion.


Definition:Depend

A statement $q$ '''depends upon''' another statement $p$ if the truth value of $q$ is ''influenced'' (in some way) by that of $p$, but is ''not necessarily'' implied by that of $p$. Some authors prefer '''rests on''' for '''depends upon'''.


Definition:Therefore

If statement $p$ logically implies statement $q$, then we may say: :'''$p$, therefore $q$'''. The symbology: :$p, q \vdash r$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' So the symbol $\vdash$ is interpreted to mean '''therefore'''. Thus, $p, q \vdash r$ reads as: :'''$p$ and $q$, therefore $r$.''' A fallacy may be indicated by $p, q \not \vdash r$, which can be interpreted as: :'''Given as premises $p$ and $q$, we may ''not'' validly conclude $r$.'''


Definition:Because

If statement $p$ logically implies statement $q$, then we may say: : '''$q$, because $p$.''' The symbol $\dashv$ is interpreted to mean '''because'''. Thus: : $r \dashv p, q$ means: :'''Given as premises $p$ and $q$, we may validly conclude $r$''' or :'''$r$, because $p$ and $q$.'''


Definition:Fallacy

A '''(logical) fallacy''' is a mistake caused by an application of an invalid argument. '''Logical fallacies''' abound. Most of the documented '''fallacies''' that can be found in the literature arise from linguistic imprecision or deliberately misleading statements, or both. Frequently the commission of '''fallacies''' is as deliberate as the telling of lies.


Definition:Sequent

A '''sequent''' is an expression in the form: :$\phi_1, \phi_2, \ldots, \phi_n \vdash \psi$ where $\phi_1, \phi_2, \ldots, \phi_n$ are premises (any number of them), and $\psi$ the conclusion (only one), of an argument.


Definition:Zermelo-Fraenkel Set Theory

'''Zermelo-Fraenkel Set Theory''' is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of set theory. These are as follows: {{:Definition:Zermelo-Fraenkel Axioms}}