hcju/proofwikips huggingface dataset:
Inverse Sine is Odd Function
Everywhere that the function is defined: :$\map \arcsin {-x} = -\arcsin x$
Arccosine of Negative Argument
Everywhere that the function is defined: :$\map \arccos {-x} = \pi - \arccos x$
Inverse Tangent is Odd Function
Everywhere that the function is defined: :$\map \arctan {-x} = -\arctan x$
Arccotangent of Negative Argument
Everywhere that the function is defined: :$\map \arccot {-x} = \pi - \arccot x$
Arcsecant of Negative Argument
Everywhere that the function is defined: :$\map \arcsec {-x} = \pi - \arcsec x$
Inverse Cosecant is Odd Function
Everywhere that the function is defined: :$\map \arccsc {-x} = -\arccsc x$
Sine of Angle of Triangle by Semiperimeter
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: : $\sin A = \dfrac 2 {b c} \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$ where $\sin$ denotes sine and $s$ is the semiperimeter: $s = \dfrac {a + b + c} 2$.
Functionally Complete Logical Connectives/Negation, Conjunction, Disjunction and Implication
: $\left\{{\neg, \land, \lor, \implies}\right\}$: Not, And, Or and Implies
Functionally Complete Logical Connectives/Negation and Disjunction
:$\set {\neg, \lor}$: Not and Or
Functionally Complete Logical Connectives/Negation and Conjunction
:$\set {\neg, \land}$: Not and And
Functionally Complete Logical Connectives/Negation and Conditional
:$\set {\neg, \implies}$: Not and Implies
Functionally Complete Logical Connectives/Conjunction, Negation and Disjunction
:$\set {\neg, \land, \lor}$: Not, And and Or
Functionally Complete Logical Connectives/NAND
:$\set {\uparrow}$: NAND
Functionally Complete Logical Connectives/NOR
:$\set {\downarrow}$: NOR
NOR with Equal Arguments/Proof 1
Let $\downarrow$ signify the NOR operation. Then for any proposition $p$: :$p \downarrow p \dashv \vdash \neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator.
NOR with Equal Arguments/Proof 2
Let $\downarrow$ signify the NOR operation. Then for any proposition $p$: :$p \downarrow p \dashv \vdash \neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator.
NAND with Equal Arguments/Proof 1
Let $\uparrow$ signify the NAND operation. Then, for any proposition $p$: :$p \uparrow p \dashv \vdash \neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation.
NAND with Equal Arguments/Proof 2
Let $\uparrow$ signify the NAND operation. Then, for any proposition $p$: :$p \uparrow p \dashv \vdash \neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation.
Conditional and Converse are not Equivalent
A conditional statement: :$p \implies q$ is not logically equivalent to its converse: :$q \implies p$
NOR is Commutative/Proof 1
Let $\downarrow$ signify the NOR operation. Then, for any two propositions $p$ and $q$: :$p \downarrow q \dashv \vdash q \downarrow p$ That is, NOR is commutative.
NOR is Commutative/Proof 2
Let $\downarrow$ signify the NOR operation. Then, for any two propositions $p$ and $q$: :$p \downarrow q \dashv \vdash q \downarrow p$ That is, NOR is commutative.
NOR is not Associative/Proof 1
Let $\downarrow$ signify the NOR operation. Then there exist propositions $p,q,r$ such that: :$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$ That is, NOR is not associative.
NOR is not Associative/Proof 2
Let $\downarrow$ signify the NOR operation. Then there exist propositions $p, q, r$ such that: :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ That is, NOR is not associative.
De Morgan's Laws (Predicate Logic)/Assertion of Universality
:$\forall x: \map P x \dashv \vdash \neg \exists x: \neg \map P x$ ::''If everything '''is''', there exists nothing that '''is not'''.''
De Morgan's Laws (Predicate Logic)/Denial of Existence
:$\forall x: \neg \map P x \dashv \vdash \neg \exists x: \map P x$ ::''If everything '''is not''', there exists nothing that '''is'''.''
De Morgan's Laws (Predicate Logic)/Denial of Universality
:$\neg \forall x: \map P x \dashv \vdash \exists x: \neg \map P x$ ::''If not everything '''is''', there exists something that '''is not'''.''
De Morgan's Laws (Predicate Logic)/Assertion of Existence
:$\neg \forall x: \neg \map P x \dashv \vdash \exists x: \map P x$ ::''If not everything '''is not''', there exists something that '''is'''.''
Conjunction in terms of NAND
:$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$ where $\land$ denotes logical conjunction and $\uparrow$ denotes logical NAND.
Disjunction in terms of NOR
:$p \lor q \dashv \vdash \paren {p \downarrow q} \downarrow \paren {p \downarrow q}$ where $\lor$ denotes logical disjunction and $\downarrow$ denotes logical NOR.
Disjunction in terms of NAND
:$p \lor q \dashv \vdash \paren {p \uparrow p} \uparrow \paren {q \uparrow q}$ where $\lor$ denotes logical disjunction and $\uparrow$ denotes logical NAND.
Implication in terms of NAND
:$p \implies q \dashv \vdash p \uparrow \paren {q \uparrow q}$
Mapping is Injection and Surjection iff Inverse is Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: : $f: S \to T$ can be defined as a bijection in the sense that: ::$(1): \quad f$ is an injection ::$(2): \quad f$ is a surjection : :the inverse $f^{-1}$ of $f$ is itself a mapping.
Condition for Composite Mapping to be Identity
Let $S$ and $T$ be sets. Let $f: S \to T$ and $g: T \to S$ be mappings such that: : $g \circ f = I_S$ where $I_S$ is the identity mapping on $S$. Then $f$ is an injection and $g$ is a surjection.
Eigenvalues of G-Representation are Roots of Unity
Let $G$ be a finite group. Let $\left({K, +, \cdot}\right)$ be a field. Let $V$ be a $G$-module over $K$ (i.e. $V$ is a $K \left[{G}\right]$-module). Then $\forall g \in G$, the eigenvalues of the action by the vector $g \in K \left[{G}\right]$ on $V$ are roots of unity.
Character of Representations over C are Algebraic Integers
Let $G$ be a finite group. Let $\chi$ be the character of any [[Definition:G-Module|$\C \left[{G}\right]$-module]] $\left({V, \rho}\right)$. Then for all $g \in G$, it follows that $\chi \left({g}\right)$ is an algebraic integer.
Injection from Finite Set to Itself is Surjection
Let $S$ be a finite set. Let $f: S \to S$ be an injection. Then $f$ is also a surjection.
Codomain of Bijection is Domain of Inverse
Let $S$ and $T$ be sets. Let $f: S \to T$ be a bijection. Let $f^{-1}: T \to S$ be the inverse of $f$. Then the domain of $f^{-1}$ equals the codomain of $f$.
Domain of Bijection is Codomain of Inverse
Let $S$ and $T$ be sets. Let $f: S \to T$ be a bijection. Let $f^{-1}: T \to S$ be the inverse of $f$. Then the codomain of $f^{-1}$ equals the domain of $f$.
Conditions for Uniqueness of Left Inverse Mapping
Let $S$ and $T$ be sets such that $S \ne \O$. Let $f: S \to T$ be an injection. Then a left inverse mapping of $f$ is in general not unique. Uniqueness occurs under either of two circumstances: :$(1): \quad S$ is a singleton :$(2): \quad f$ is a bijection.
Surjection iff Right Inverse/Non-Uniqueness
A right inverse of $f$ is in general not unique. Uniqueness occurs  $f$ is a bijection.
Mapping reflects Preordering
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let ${\precsim} \subseteq T \times T$ be a preordering on $T$. Let $\RR$ be the relation defined on $S$ by the rule: :$x \mathrel \RR y \iff \map f x \precsim \map f y$ Then $\RR$ is a preordering on $S$.
Bernoulli's Inequality
Let $x \in \R$ be a real number such that $x > -1$. Let $n \in \Z_{\ge 0}$ be a positive integer. Then: :$\paren {1 + x}^n \ge 1 + n x$
Multiple of Ring Product
Let $\left({R, +, \circ}\right)$ be a ring. Let $x, y \in \left({R, +, \circ}\right)$. Then: :$\forall n \in \Z_{> 0}: \left({n \cdot x} \right) \circ y = n \cdot \left({x \circ y}\right) = x \circ \left({n \cdot y}\right)$ where $n \cdot x$ denotes the $n$th multiple of $x$.
Ordering is Preserved on Integers by Addition
The usual ordering on the integers is preserved by the operation of addition: :$\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$
Power Set of Finite Set is Finite
Let $S$ be a finite set. Then the power set of $S$ is likewise finite.
Common Divisor Divides Difference
Let $c$ be a common divisor of two integers $a$ and $b$. That is: :$a, b, c \in \Z: c \divides a \land c \divides b$ Then: :$c \divides \paren {a - b}$
Divisor Divides Multiple
Let $a, b$ be integers. Let: :$a \divides b$ where $\divides$ denotes divisibility. Then: :$\forall c \in \Z: a \divides b c$
One is not Prime
The integer $1$ (one) is not a prime number.
Initial Part of String is Substring
Let $S$ be a string, and let $T$ be an initial part of $S$. Then $T$ is a substring of $S$.
Ring Zero is Idempotent
Let $\left({R, +, \circ}\right)$ be a ring whose ring zero is $0_R$. Then $0_R$ is an idempotent element of $R$ under the ring product $\circ$: :$0_R \circ 0_R = 0_R$
Prime iff Coprime to all Smaller Positive Integers
Let $p$ be a prime number Then: :$\forall x \in \Z, 0 < x < p: x \perp p$ That is, $p$ is relatively prime to all smaller (strictly) positive integers.
Division Theorem/Positive Divisor/Positive Dividend
 For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$  In the above equation: * $a$ is the '''dividend''' * $b$ is the '''divisor''' * $q$ is the '''quotient''' * $r$ is the '''principal remainder''', or, more usually, just the '''remainder'''.
Division Theorem/Positive Divisor
 For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$  In the above equation: :$a$ is the '''dividend''' :$b$ is the '''divisor''' :$q$ is the '''quotient''' :$r$ is the '''principal remainder''', or, more usually, just the '''remainder'''.
Division Theorem/Positive Divisor/Positive Dividend/Existence
For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Positive Dividend/Uniqueness
For every pair of integers $a, b$ where $a \ge 0$ and $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, a \ge 0, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Existence/Proof 2
For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Existence
For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Uniqueness/Proof 1
For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Uniqueness/Proof 2
For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Uniqueness
For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Existence/Proof 3
For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$: :$\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$
Division Theorem/Positive Divisor/Uniqueness/Proof 3
For every pair of integers $a, b$ where $b > 0$, the integers $q, r$ such that $a = q b + r$ and $0 \le r < b$ are unique: :$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
Integer Coprime to all Factors is Coprime to Whole
Let $a, b \in \Z$. Let $\displaystyle b = \prod_{j \mathop = 1}^r b_j$ Let $a$ be coprime to each of $b_1, \ldots, b_r$. Then $a$ is coprime to $b$.
All Factors Divide Integer then Whole Divides Integer
Let $S = \set {a_1, a_2, \ldots, a_r} \subseteq \Z$ be a finite subset of the integers. Let $S$ be pairwise coprime. Let: :$\forall j \in \set {1, 2, \ldots, r}: a_r \divides b$ where $\divides$ denotes divisibility. Then: :$\displaystyle \prod_{j \mathop = 1}^r a_j \divides b$
Piecewise Continuous Function with One-Sided Limits is Bounded
Let $f$ be a real function defined on a closed interval $\closedint a b$. Let $f$ be a piecewise continuous function with one-sided limits. Then $f$ is a bounded piecewise continuous function.
Upper Bounds for Prime Numbers/Result 1
:$\forall n \in \N: \map p n \le 2^{2^{n - 1} }$
Upper Bounds for Prime Numbers/Result 2
:$\forall n \in \N: \map p n \le \paren {p \paren {n - 1} }^{n - 1} + 1$
Product of Integers of form 4n + 1
Let $m, n \in \Z$ such that both $m$ and $n$ are of the form $4 k + 1$ where $k \in \Z$. Then $m n$ is also of the form $4 k + 1$.
Infinite Number of Primes of form 4n - 1
There are infinitely many prime numbers of the form $4 n - 1$.
Integer Coprime to Modulus iff Linear Congruence to 1 exists
Let $a, m \in \Z$. The linear congruence: :$a x \equiv 1 \pmod m$ has a solution $x$  $a$ and $m$ are coprime.
Integer Coprime to Modulus iff Linear Congruence to 1 exists/Corollary
Let $p$ be a prime number. The linear congruence: :$a x \equiv 1 \pmod p$ has a solution $x$  $a \not \equiv 0 \pmod p$.
Solution of Linear Congruence/Existence
$a x \equiv b \pmod n$ has at least one solution : : $\gcd \set {a, n} \divides b$ that is,  $\gcd \set {a, n}$ is a divisor of $b$.
Solution of Linear Congruence/Number of Solutions
Let $\gcd \set {a, n} = d$. Then $a x \equiv b \pmod n$ has $d$ solutions which are given by the unique solution modulo $\dfrac n d$ of the congruence: : $\dfrac a d x \equiv \dfrac b d \paren {\bmod \dfrac n d}$
Solution of Linear Congruence/Unique iff Coprime to Modulus
If $\gcd \set {a, n} = 1$, then $a x \equiv b \pmod n$ has a unique solution.
Ring Direct Product of Modulo Integers is Isomorphic to Ring Modulo Product iff Coprime
Let $m, n \in \Z_{>1}$. Let $\struct {\Z_m, +_m, \times_m}$ and $\struct {\Z_n, +_n, \times_n}$ be the rings of integers modulo $m$ and $n$ respectively. Let $\struct {\Z_m \times \Z_n}$ be the direct product of $\Z_m$ and $\Z_n$. Let $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$ be the ring of integers modulo $mn$. Then $\struct {\Z_m \times \Z_n}$ is isomorphic to $\struct {\Z_{m n}, +_{m n}, \times_{m n} }$  $m$ and $n$ are coprime.
Trivial Norm on Division Ring is Non-Archimedean
Let $\struct {R, +, \circ}$ be a division ring whose ring zero is $0_R$. Then the trivial norm $\norm {\, \cdot \,}: R \to \R_{\ge 0}$, which is given by: :$\norm x = \begin{cases}   0 & : x = 0_R \\   1 & : \text{ otherwise} \end{cases}$ is non-archimedean: :$\norm {x + y} \le \max \set {\norm x, \norm y}$
Ring of Integers Modulo m is Ring
For all $m \in \N: m \ge 2$, the ring of integers modulo $m$: :$\struct {\Z_m, +_m, \times_m}$ is a commutative ring with unity $\eqclass 1 m$. The zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.
Piecewise Continuous Function with One-Sided Limits is Uniformly Continuous on Each Piece
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $f$ be piecewise continuous with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}} Then: :for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is uniformly continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$ .
Principle of Definition by Structural Induction
Let $\LL$ be a formal language. Let the formal grammar of $\LL$ be a bottom-up grammar with unique parsability. A definition $\map D \phi$ (in the metalanguage of $\LL$) for all well-formed formulas $\phi$ of $\LL$ is uniquely specified by: :$(1): \quad$ A definition $\map D a$ for each letter $a$ of $\LL$ :$(2): \quad$ For each rule of formation for $\LL$, a definition $\map D \phi$ of the resultant WFF $\phi$ in terms of the constituent WFFs' definitions $\map D {\phi_1}, \ldots, \map D {\phi_n}$.
Principle of Structural Induction
Let $\LL$ be a formal language. Let the formal grammar of $\LL$ be a bottom-up grammar. Let $\map P \phi$ be a statement (in the metalanguage of $\LL$) about well-formed formulas $\phi$ of $\LL$. Then $P$ is true for all WFFs of $\LL$  both: :$\map P a$ is true for all letters $a$ of $\LL$, and, for each rule of formation of $\LL$, if $\phi$ is a WFF resulting from WFFs $\phi_1, \ldots, \phi_n$ by applying that rule, then: :$\map P \phi$ is true only if $\map P {\phi_1}, \ldots, \map P {\phi_n}$ are all true.
Integers under Multiplication form Monoid
The set of integers under multiplication $\struct {\Z, \times}$ is a monoid.
Integers under Multiplication form Semigroup
The set of integers under multiplication $\struct {\Z, \times}$ is a semigroup.
Canonical Form of Rational Number is Unique
The canonical form of a rational number is unique.
Existence of Canonical Form of Rational Number
Let $r \in \Q$. Then: :$\exists p \in \Z, q \in \Z_{>0}: r = \dfrac p q, p \perp q$ That is, every rational number can be expressed in its canonical form.
Existence of Dyadic Rational between two Rationals
Let $a$ and $b$ be rational numbers such that $a < b$. Then there exist integers $m$ and $r$ such that: :$a < \dfrac m {2^r} < b$ That is, there exists a dyadic rational between any pair of rational numbers.
Field with 4 Elements has only Order 2 Elements
Let $\struct {\GF, +, \times}$ be a field which has exactly $4$ elements. Then: :$\forall a \in \GF: a + a = 0_\GF$ where $0_\GF$ is the zero of $\GF$.
Count of Truth Functions
There are $2^{\paren {2^k} }$ distinct truth functions on $k$ variables.
Unary Truth Functions
There are $4$ distinct unary truth functions: :$(1): \quad$ The constant function $\map f p = \F$ :$(2): \quad$ The constant function $\map f p = \T$ :$(3): \quad$ The identity function $\map f p = p$ :$(4): \quad$ The logical not function $\map f p = \neg p$
Binary Truth Functions
There are $16$ distinct binary truth functions: * Two constant functions: ** $\map {f_\F} {p, q} = \F$ ** $\map {f_\T} {p, q} = \T$ * Two projections: ** $\map {\pr_1} {p, q} = p$ ** $\map {\pr_2} {p, q} = q$ * Two negated projections: ** $\map {\overline {\pr_1} } {p, q} = \neg p$ ** $\map {\overline {\pr_2} } {p, q} = \neg q$ * The conjunction: $p \land q$ * The disjunction: $p \lor q$ * Two conditionals: ** $p \implies q$ ** $q \implies p$ * The biconditional: $p \iff q$ * The exclusive or: $\map \neg {p \iff q}$ * Two negated conditionals: ** $\map \neg {p \implies q}$ ** $\map \neg {q \implies p}$ * The NAND: $p \uparrow q$ * The NOR: $p \downarrow q$
Inverse Hyperbolic Sine Logarithmic Formulation
{{finish|The true story is more complicated than this. The inverse hyperbolic sine is a multifunction. So is $\ln$ in the complex plane. And $\sqrt {z^2 + 1}$ has two values, pos and neg, so you need to justify which one is taken. The best context to put this page is probably into Definition:Inverse Hyperbolic Sine/Arcsine, where an explanation needs to be made as to why the principal value is taken thus.}} For any complex number $z \in \C$: :$\operatorname {arsinh} z = \map \ln {z + \sqrt {z^2 + 1} }$ where $\operatorname {arsinh} z$ is the inverse hyperbolic sine.
Value of Cauchy Determinant
Let $D_n$ be a Cauchy determinant of order $n$: :$\begin{vmatrix} \dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{vmatrix}$ Then the value of $D_n$ is given by: :$D_n = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_j - y_i}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i + y_j}\right)}$ If $D_n$ is given by: :$\begin{vmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{vmatrix}$ then its determinant is given by: :$D_n = \dfrac {\displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \left({x_j - x_i}\right) \left({y_i - y_j}\right)} {\displaystyle \prod_{1 \mathop \le i, \, j \mathop \le n} \left({x_i - y_j}\right)}$
Integers under Addition form Abelian Group
The set of integers under addition $\struct {\Z, +}$ forms an abelian group.
Piecewise Continuous Function with One-Sided Limits is Darboux Integrable
Let $f$ be a real function defined on a closed interval $\closedint a b$. Let $f$ be piecewise continuous with one-sided limits on $\closedint a b$. Then $f$ is Darboux integrable on $\closedint a b$.
Schröder Rule/Proof 1
Let $A$, $B$ and $C$ be relations on a set $S$. Then the following are equivalent statements: :$(1): \quad A \circ B \subseteq C$ :$(2): \quad A^{-1} \circ \overline{C} \subseteq \overline{B}$ :$(3): \quad \overline{C} \circ B^{-1} \subseteq \overline{A}$ where: : $\circ$ denotes relation composition : $A^{-1}$ denotes the inverse of $A$ : $\overline{A}$ denotes the complement of $A$.
Condition for Well-Foundedness/Forward Implication
Let $\struct {S, \preceq}$ be an ordered set. Let $\struct {S, \preceq}$ be well-founded. Then there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that $\forall n \in \N: a_{n + 1} \prec a_n$.
Extendability Theorem for Function Continuous on Open Interval
Let $f$ be a continuous real function that is defined on an open interval $\openint a b$. Let $g$ be a real function that satisfies: :$g$ is defined on $\closedint a b$ :$g$ is continuous on $\closedint a b$ :$g = f$ on $\openint a b$. Then $g$ exists  $\displaystyle \lim_{x \mathop \to a^+} \map f x$ and $\displaystyle \lim_{x \mathop \to b^-} \map f x$ exist.
Definition:Trivial Ordering
The '''trivial ordering''' is an ordering $\mathcal R$ defined on a set $S$ by: :$\forall a, b \in S: a \mathrel{\mathcal R} b \iff a = b$ That is, there is no ordering defined on any two distinct elements of the set $S$.
Integrability Theorem for Functions Continuous on Open Intervals
Let $f$ be a real function defined on a closed interval $\closedint a b$ such that $a < b$. Let $f$ be continuous on $\openint a b$. Let the one-sided limits $\displaystyle \lim_{x \mathop \to a^+} \map f x$ and $\displaystyle \lim_{x \mathop \to b^-} \map f x$ exist. Then $f$ is Darboux integrable on $\closedint a b$.
Set of Rational Numbers whose Numerator Divisible by p is Closed under Multiplication
Let $p$ be a prime number. Let $A_p$ be the set of all rational numbers which, when expressed in canonical form has a numerator which is divisible by $p$. Then $A_p$ is closed under rational multiplication.
Integer Less One divides Power Less One
Let $q, n \in \Z_{>0}$. Then: :$\paren {q - 1} \divides \paren {q^n - 1}$ where $\divides$ denotes divisibility.
Number Plus One divides Power Plus One iff Odd
Let $q, n \in \Z_{>0}$. Then: :$\paren {q + 1} \divides \paren {q^n + 1}$  $n$ is odd. In the above, $\divides$ denotes divisibility.
Mapping Bounded on Union iff Bounded on Each Component/Real-Valued Function
Let $f$ be a real-valued function. Then: : $f$ is bounded on the union of a finite number of sets within the domain of $f$ : :$f$ is bounded on each of the sets.