hcju/proofwikips huggingface dataset:
Sum of All Ring Products is Closed under Addition
Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +}\right)$ and $\left({T, +}\right)$ be additive subgroups of $\left({R, +, \circ}\right)$. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\ n}\right]}\right\}$ Then $\left({S T, +}\right)$ is a closed subset of $\left({R, +}\right)$.
Complex Numbers form Subfield of Quaternions
The field of complex numbers $\left({\C, +, \times}\right)$ is isomorphic to the subfields of the quaternions $\left({\mathbb H, +, \times}\right)$ whose underlying subsets are: :$(1): \quad \mathbb H_\mathbf i = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: c = d = 0}\right\}$ :$(2): \quad \mathbb H_\mathbf j = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: b = d = 0}\right\}$ :$(3): \quad \mathbb H_\mathbf k = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: b = c = 0}\right\}$ That is: :$(1): \quad \mathbb H_\mathbf i = \left\{{a \mathbf 1 + b \mathbf i \in \mathbb H}\right\}$ :$(2): \quad \mathbb H_\mathbf j = \left\{{a \mathbf 1 + c \mathbf j \in \mathbb H}\right\}$ :$(3): \quad \mathbb H_\mathbf k = \left\{{a \mathbf 1 + d \mathbf k \in \mathbb H}\right\}$
Ring Homomorphism whose Kernel contains Ideal
Let $R$ be a ring. Let $J$ be an ideal of $R$. Let $\nu: R \to R / J$ be the quotient epimorphism. Let $\phi: R \to S$ be a ring homomorphism such that: :$J \subseteq \map \ker \phi$ where $\map \ker \phi$ is the kernel of $\phi$. Then there exists a unique ring homomorphism $\psi: R / J \to S$ such that: :$\phi = \psi \circ \nu$ where $\circ$ denotes composition of mappings. :300px Also: :$\map \ker \psi = \map \ker \phi / J$
Decay Equation
The first order ordinary differential equation: :$\dfrac {\d y} {\d x} = k \paren {y_a - y}$ where $k \in \R: k > 0$ has the general solution: :$y = y_a + C e^{-k x}$ where $C$ is an arbitrary constant. If $y = y_0$ at $x = 0$, then: :$y = y_a + \paren {y_0 - y_a} e^{-k x}$ This differential equation is known as the '''decay equation'''.
Relation on Empty Set is Equivalence
Let $S = \varnothing$, that is, the empty set. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is the null relation and is an equivalence relation.
Increasing Union of Subrings is Subring
Let $R$ be a ring. Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ be subrings of $R$. Then the increasing union $S$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ is a subring of $R$.
Subrings of Integers are Sets of Integer Multiples
Let $\struct {\Z, +, \times}$ be the integral domain of integers. The subrings of $\struct {\Z, +, \times}$ are the rings of integer multiples: :$\struct {n \Z, +, \times}$ where $n \in \Z: n \ge 0$. There are no other subrings of $\struct {\Z, +, \times}$ but these.
Subring of Integers is Ideal
Every subring of $\struct {\Z, +, \times}$ is an ideal of the ring $\struct {\Z, +, \times}$.
Polynomial Ring of Sequences is Ring
Let $R$ be a ring. Let $P \left[{R}\right]$ be the polynomial ring over sequences in $R$. Then $P \left[{R}\right]$ is itself a ring.
Integer Addition is Closed
The set of integers is closed under addition: :$\forall a, b \in \Z: a + b \in \Z$
Integer Addition Identity is Zero
The identity of integer addition is $0$: :$\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$
Inverses for Integer Addition
Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition: :$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$
Rational Numbers form Ring
The set of rational numbers $\Q$ forms a ring under addition and multiplication: $\struct {\Q, +, \times}$.
Real Numbers form Ring
The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.
Complex Numbers form Ring
The set of complex numbers $\C$ forms a ring under addition and multiplication: $\struct {\C, +, \times}$.
Complex Addition Identity is Zero
Let $\C$ be the set of complex numbers. The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$.
Rational Addition Identity is Zero
The identity of rational number addition is $0$: :$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$
Inverse for Complex Addition
Let $z = x + i y \in \C$ be a complex number. Let $-z = -x - i y \in \C$ be the negative of $z$. Then $-z$ is the inverse element of $z$ under the operation of complex addition: :$\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$
Equal Elements of Field of Quotients
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {K, +, \circ}$ be the field of quotients of $\struct {D, +, \circ}$. Let $x = \dfrac p q \in K$. Then: :$\forall k \in D^*: x = \dfrac {p \circ k} {q \circ k}$ where: :$D^* := D \setminus \set {0_D}$ that is, $D$ with its zero removed.
Rational Addition is Closed
The operation of addition on the set of rational numbers $\Q$ is well-defined and closed: :$\forall x, y \in \Q: x + y \in \Q$
Rational Multiplication is Closed
The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed: :$\forall x, y \in \Q: x \times y \in \Q$
Rational Multiplication Identity is One
The identity of rational number multiplication is $1$: :$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$
Inverses for Rational Multiplication
Each element $x$ of the set of non-zero rational numbers $\Q^*$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication: :$\displaystyle \forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$
Real Addition Identity is Zero
The identity of real number addition is $0$: :$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$
Inverses for Real Addition
Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition: :$\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$
Real Multiplication Identity is One
The identity element of real number multiplication is the real number $1$: :$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$
Inverses for Real Multiplication
Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication: :$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$
Complex Multiplication Identity is One
Let $\C_{\ne 0}$ be the set of complex numbers without zero. The identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.
Inverse for Complex Multiplication
Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $\dfrac 1 z$ under the operation of complex multiplication: :$\forall z \in \C_{\ne 0}: \exists \dfrac 1 z \in \C_{\ne 0}: z \times \dfrac 1 z = 1 + 0 i = \dfrac 1 z \times z$ This inverse can be expressed as: :$\dfrac 1 z = \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$ where $\overline z$ is the complex conjugate of $z$.
Additive Group of Integers is Normal Subgroup of Complex
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$.
Rational Numbers form Subfield of Complex Numbers
Let $\struct {\Q, +, \times}$ denote the field of rational numbers. Let $\struct {\C, +, \times}$ denote the field of complex numbers. $\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$.
Subfield of Subfield is Subfield
Let $R$ be a ring with unity. Let $K_1, K_2$ be fields, such that: : $K_1$ is a subfield of $R$ : $K_2$ is a subfield of $K_1$ Then $K_2$ is a subfield of $R$.
Characterisation of Jacobson Radical
Let $A$ be a commutative ring with unity. Let $A^\times$ be the group of units of $A$. Let $\map {\operatorname {Jac} } A$ be the Jacobson radical of $A$. Then: :$\map {\operatorname {Jac} } A = \set {a \in A: 1_A - a x \in A^\times \text{ for all } x \in A}$ where $1_A$ is the unity of $A$.
Prime Ideal iff Quotient Ring is Integral Domain
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a prime ideal of $R$  the quotient ring $R / J$ is an integral domain.
Maximal Ideal of Commutative and Unitary Ring is Prime Ideal
Let $R$ be a commutative ring with unity. Let $M$ be a maximal ideal of $R$. Then $M$ is a prime ideal of $R$.
Radical Ideal iff Quotient Ring is Reduced
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity. Let $J$ be an ideal of $R$. Then $J$ is a radical ideal  the quotient ring $R / J$ is a reduced ring.
Properties of Degree
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$. Let $R \sqbrk X$ denote the ring of polynomial forms over $R$ in the indeterminate $X$. For $f \in R \sqbrk X$ let $\map \deg f$ denote the degree of $f$. Then the following hold:
Set of Monomials is Closed Under Multiplication
Let $M$ be the set of all monomials on the set $\set {X_j: j \in J}$, with multiplication $\circ$ defined by: :$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$ Then $M$ is closed under $\circ$.
Polynomial Addition is Commutative
Addition of polynomials is commutative.
Multiplication of Polynomials is Associative
Multiplication of polynomials is associative.
Polynomials Contain Multiplicative Identity
The set of polynomials has a multiplicative identity.
Multiplication of Polynomials is Commutative
Multiplication of polynomials is commutative.
Multiplication of Polynomials Distributes over Addition
Multiplication of polynomials is left- and right- distributive over addition.
Distributivity is Preserved in Induced Structure
Let $\struct {T, \oplus, \otimes}$ be an algebraic structure, and let $S$ be a set. Let $T^S$ denote the set of all mappings from $S$ to $T$. Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\oplus$. Let $\struct {T^S, \otimes}$ be the structure on $T^S$ induced by $\otimes$. If $\otimes$ is distributive over $\oplus$, then the pointwise operation induced on $T^S$ by $\otimes$ is distributive over the operation induced by $\oplus$.
Dirichlet Series Absolute Convergence Lemma
Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series. Suppose that $f$ converges absolutely at $s_0 = \sigma_0 + i t_0 \in \C$. Then $f$ converges absolutely at all points $s = \sigma + i t \in \C$ with $\sigma \ge \sigma_0$.
Existence of Abscissa of Absolute Convergence
Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \left\vert { a_n n^{-s} } \right\vert$ not converge for all $s \in \C$, or diverge for all $s \in \C$. Then there exists a real number $\sigma_a$ such that $f \left({s}\right)$ converges absolutely for all $s = \sigma + it$ with $\sigma > \sigma_a$, and does not converge absolutely for all $s$ with $\sigma < \sigma_a$. We call $\sigma_a$ the '''abscissa of absolute convergence''' of the Dirichlet series.
Dirichlet Series Convergence Lemma
Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series. Let $f \left({s}\right)$ converge at $s_0 = \sigma_0 + i t_0$. Then $f \left({s}\right)$ converge for all $s = \sigma + i t$ where $\sigma > \sigma_0$.
Existence of Abscissa of Convergence/General
Let $s = \sigma + i t$ Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$ be a general Dirichlet series. Then there exists a extended real number, $\sigma_0$, such that :$(1): \quad$ For $\sigma < \sigma_0$, $\map f s$ diverges :$(2): \quad$ For $\sigma > \sigma_0$, $\map f s$ converges.
Difference of Abscissae of Convergence
Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Suppose that $\map f s$ has finite Abscissa of Convergence $\sigma_c$. Then the Abscissa of Absolute Convergence $\sigma_a$ is finite, and: :$0 \le \sigma_a - \sigma_c \le 1$
Completion Theorem (Metric Space)
Let $M = \struct {A, d}$ be a metric space. Then there exists a completion $\tilde M = \struct {\tilde A, \tilde d}$ of $\struct {A, d}$. Moreover, this completion is unique up to isometry. That is, if $\struct {\hat A, \hat d}$ is another completion of $\struct {A, d}$, then there is a bijection $\tau: \tilde A \leftrightarrow \hat A$ such that: :$(1): \quad \tau$ restricts to the identity on $x$: :::: $\forall x \in A: \map \tau x = x$ :$(2): \quad \tau$ preserves metrics: :::: $\forall x_1, x_2 \in A : \map {\hat d} {\map \tau {x_1}, \map \tau {x_2} } = \map {\tilde d} {x_1, x_2}$
Completeness Criterion (Metric Spaces)
Let $M = \struct {S, d}$ be a metric space. Let $A \subseteq S$ be a dense subset. Suppose that every Cauchy sequence in $A$ converges in $M$. Then $M$ is complete.
Natural Number Addition is Closed
The operation of addition on the set of natural numbers $\N$ is closed: :$\forall x, y \in \N: x + y \in \N$
Properties of Ordered Field
Let $\struct {F, +, \cdot}$ be an ordered field with unity $1$, zero $0$. Denote the strict order by $<$ and the weak order by $\le$. Let $\Char F$ denote the characteristic of $F$. Then the following hold for all $x, y, z \in k$: :$(1): \quad x < 0 \iff -x > 0$ :$(2): \quad x > y \iff x-y > 0$ :$(3): \quad x < y \iff -x > -y$ :$(4): \quad \paren {z < 0} \land \paren {x < y} \implies x z > y z$ :$(5): \quad x \ne 0 \implies x^2 > 0$ :$(6): \quad 1 > 0$ :$(7): \quad \Char k = 0$ :$(8): \quad x > y > 0 \iff y^{-1} > x^{-1} > 0$
Characterisation of Ordered Fields
Let $\struct {k, +, \cdot}$ be a field with unity $1$ and zero $0$. Then the following are equivalent: :$(1): \quad$ There exists a total ordering $\le$ on $k$ such that $\struct {k, \le}$ is an ordered field :$(2): \quad$ $-1$ cannot be written as a sum of squares of elements of $k$ :$(3): \quad$ $0$ cannot be written as a non-empty sum of squares of non-zero elements of $k$
Hahn-Banach Theorem
Let $E$ be a vector space over $\R$. Let $p: E \to \R$ be a Minkowski functional. Let $G \subseteq E$ be a linear subspace of $E$. Let $f: G \to \R$ be a linear functional such that: :$\forall x \in G: \map f x \le \map p x$ Then there exists a linear functional $\tilde f$ defined on the whole space $E$ which extends $f$. That is: :$\forall x \in G: \map {\tilde f} x = \map f x$ such that: :$\forall x \in E: \map {\tilde f} x \le \map p x$
Hardy-Littlewood Circle Method
Let $\AA$ be a subset of the non-negative integers. Let: :$\displaystyle \map T s = \sum_{a \mathop \in \AA} s^a$ be the generating function for $\AA$. For $N \in \N$, let $\map {r_{\AA, \ell} } N$ be the number of solutions $\tuple {x_1, \ldots, x_\ell} \in \AA^\ell$ to the equation: :$x_1 + \cdots + x_\ell = N$ Then: :$\displaystyle \forall \rho \in \openint 0 1: \map {r_{\AA, \ell} } N = \oint_{\size s \mathop = \rho} \frac {\map T s^\ell} {s^{N + 1} } \rd s$
Vinogradov Circle Method
$\newcommand{\A}{\mathcal A}$Let $\A$ be a subset of the non-negative integers. For $\alpha \in \R$, let: :$e \left({\alpha}\right) := \exp \left({2 \pi i \alpha} \right)$ Let: :$\displaystyle T_N \left({s}\right) = \sum_{\substack {a \mathop \in \A \\ a \mathop \le N}} s^a$ be the truncated generating function for $\A$. {{explain|"truncated" generating function}} Let: :$V_N \left({\alpha}\right) := T_N \left({e \left({\alpha}\right)}\right)$ Let $r_{\A, \ell} \left({N}\right)$ be the number of solutions $\left({x_1, \dotsc, x_\ell}\right) \in \A^\ell$ to the equation: :$(1): \quad \displaystyle x_1 + \cdots + x_l = N$ Then: :$\displaystyle r_{\A, \ell} \left({N}\right) = \int_0^1 V_N \left({\alpha}\right)^\ell e \left({-N \alpha}\right) \, \mathrm d \alpha$
Euler Product
Let $a_n : \N \to \C$ be an arithmetic function. Let $\displaystyle f \left({s}\right) = \sum_{n \mathop \in \N} a_n n^{-s}$ be its Dirichlet series. Let $\sigma_a$ be its abscissa of absolute convergence. Then for $\Re \left({s}\right) > \sigma_a$: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-s} = \prod_p \frac 1 {1 - a_p p^{-s} }$ where $p$ ranges over the primes. This representation for $f$ is called an '''Euler product''' for the Dirichlet series. {{stub|Completely multiplicative hypothesis not mentioned. Needs also the statement: $\displaystyle f \left({s}\right) {{=}} \prod_p \left\{{\sum_{k \mathop \ge 1} a_{p^k} p^{-k s}}\right\}$ or however it goes for multiplicative functions which are not completely multiplicative}}
Correspondence between Linear Group Actions and Linear Representations
Let $\left({k, +, \cdot}\right)$ be a field. Let $V$ be a vector space over $k$ of finite dimension. Let $\left({G, *}\right)$ be a finite group. There is a one-to-one correspondence between linear group actions of $G$ on $V$ and linear representations of $G$ in $V$, as follows: Let $\phi : G \times V \to V$ be a group action. Let $\rho : G \to \operatorname{Sym}(V)$ be a permutation representation of $G$ on $V$. The following are equivalent: :$(1): \quad$ $\rho$ is the permutation representation associated to $\phi$ :$(2): \quad$ $\phi$ is the group action associated to $\rho$ If this is the case, the following are equivalent: :$(1): \quad$ $\rho$ is a linear representation :$(2): \quad$ $\phi$ is a linear group action
Set Difference with Disjoint Set
Let $S, T$ be sets. Then: :$S \cap T = \O \iff S \setminus T = S$ where: :$S \cap T$ denotes set intersection :$\O$ denotes the empty set :$S \setminus T$ denotes set difference.
Integer Subtraction is Closed
The set of integers is closed under subtraction: :$\forall a, b \in \Z: a - b \in \Z$
Rational Subtraction is Closed
The set of rational numbers is closed under subtraction: :$\forall a, b \in \Q: a - b \in \Q$
Real Subtraction is Closed
The set of real numbers is closed under subtraction: :$\forall a, b \in \R: a - b \in \R$
Complex Subtraction is Closed
The set of complex numbers is closed under subtraction: :$\forall a, b \in \C: a - b \in \C$
Positive Real Numbers Closed under Division
The set $\R_{>0}$ of strictly positive real numbers is closed under division: :$\forall a, b \in \R_{>0}: a \div b \in \R_{>0}$
Rational Numbers form Integral Domain
The set of rational numbers $\Q$ forms an integral domain under addition and multiplication: $\struct {\Q, +, \times}$.
Real Numbers form Integral Domain
The set of real numbers $\R$ forms an integral domain under addition and multiplication: $\struct {\R, +, \times}$.
Properties of Content
Let $f, g \in \Q \sqbrk X$ and $q \in \Q$. The content of a polynomial satisfies the following:
Complex Numbers form Integral Domain
The set of complex numbers $\C$ forms an integral domain under addition and multiplication: $\struct {\C, +, \times}$.
Root of Polynomial iff Divisible by Minimal Polynomial
Let $K$ be a field. Let $L / K$ be a field extension of $K$. Let $\alpha \in L$ be algebraic over $K$. Then there is a unique monic polynomial $\mu_\alpha \in K \left[{X}\right]$ of least degree such that $\mu_\alpha \left({\alpha}\right) = 0$. Moreover $f \in K \left[{X}\right]$ is such that $f \left({\alpha}\right) = 0$  $\mu_\alpha$ divides $f$. {{explain|Link to an appropriate page defining divisibility among polynomials.}}
Gauss's Lemma on Unique Factorization Domains
Let $R$ be a unique factorization domain. Then the ring of polynomials $R \sqbrk X$ is also a unique factorization domain.
Gröbner Basis
Let $F$ be a finite set of polynomials. {{explain|Explain what the underlying field of the polynomials is. Real numbers, abstract rings, general fields? Link to the appropriate definition for the case.}} Let $SP \left({f_1, f_2}\right)$ be the $S$-polynomial of $f_1$ and $f_2$. Let $g$ be a polynomial. Let $RF \left({g}\right)$ be the reduced form of $g$. Then $F$ is a Gröbner basis : :$\forall f_1, f_2 \in F \left({RF \left({F, SP \left({f_1, f_2}\right)}\right) = 0}\right)$ {{explain|The meaning of the above needs to be made clear. Presumably it should be interpreted something like: $\forall f_1, f_2 \in F: RF \left({F, SP \left({f_1, f_2}\right)}\right) {{=}} 0$. But further to this: $RF$ as defined takes a single argument, a polynomial (still to be defined upon what domain). As this statement stands, $RF$ seems to take two parameters: $F$ and $SP$. Clarification is needed.}}
Numbers of Type Integer a plus b root 2 form Subdomain of Reals
Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ That is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, form an integral subdomain of the real numbers $\R$.
Sufficient Conditions for Uncountability
Let $X$ be a set. The following are equivalent: :$(1): \quad X$ contains an uncountable subset :$(2): \quad X$ is uncountable :$(3): \quad $ Every sequence of distinct points $\sequence {x_n}_{n \mathop \in \N}$ in $X$ omits at least one $x \in X$ :$(4): \quad $ There is no surjection $\N \twoheadrightarrow X$ :$(5): \quad X$ is infinite and there is no bijection $X \leftrightarrow \N$ Assuming the Continuum Hypothesis holds, we also have the equivalent uncountability condition: :$(6): \quad $There exist extended real numbers $a < b$ and a surjection $X \to \closedint a b$
Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$. Then: :$\forall x \in D: x \ne 0_D \iff \map P {x \times x}$ where $\map P {x \times x}$ denotes that $x \times x$ has the (strict) positivity property. That is, the square of any element of an ordered integral domain is (strictly) positive  that element is non-zero.
Simultaneous Equation With Two Unknowns
A pair of simultaneous linear equations of the form: {{begin-eqn}} {{eqn | l=a x + b y       | r=c }} {{eqn | l=d x + e y       | r=f }} {{end-eqn}} where $ ae \ne b d$, has as its only solution: {{begin-eqn}} {{eqn | l=x       | r= \frac {c e - b f} {a e - b d} }} {{eqn | l=y       | r=\frac {a f - c d} {a e - b d} }} {{end-eqn}}
Open Sets in Real Number Line
Every non-empty open set $I \subseteq \R$ can be expressed as a countable union of pairwise disjoint open intervals. If: :$\displaystyle I = \bigcup_{n \mathop \in \N} J_n$ :$\displaystyle I = \bigcup_{n \mathop \in \N} K_n$ are two such expressions, then there exists a permutation $\sigma$ of $\N$ such that: :$\forall n \in \N: J_n = K_{\map \sigma n}$
Unity of Ordered Integral Domain is Strictly Positive
Let $\struct {D, +, \times \le}$ be an ordered integral domain whose unity is $1_D$. Then: :$\map P {1_D}$ where $P$ is the (strict) positivity property.
Integers form Ordered Integral Domain
The integers $\Z$ form an ordered integral domain under addition and multiplication.
Rational Numbers form Ordered Integral Domain
The rational numbers $\Q$ form an ordered integral domain under addition and multiplication.
Real Numbers form Ordered Integral Domain
The set of real numbers $\R$ forms an ordered integral domain under addition and multiplication: $\struct {\R, +, \times, \le}$.
General Positivity Rule in Ordered Integral Domain
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$. Let $S \subset D$ be a subset of $D$ such that: :$\forall s \in S: \map P x$ Then the following are true: :$\displaystyle \forall n \in \N_{>0}: \forall s_i \in S: \map P {\sum_{i \mathop = 1}^n s_i}$ :$\displaystyle \forall n \in \N_{>0}: \forall s_i \in S: \map P {\prod_{i \mathop = 1}^n s_i}$ where: :$\displaystyle \sum_{i \mathop = 1}^n s_i = s_1 + s_2 + \cdots + s_n$ :$\displaystyle \prod_{i \mathop = 1}^n s_i = s_1 \times s_2 \times \cdots \times s_n$ That is, the ring sum and ring product of any number of elements in $D$ which have the (strict) positivity property also have the (strict) positivity property.