hcju/proofwikips huggingface dataset:
Definition:Pointwise Inequality of Extended Real-Valued Functions
Let $S$ be a set, and let $f,g: S \to \overline{\R}$ be extended real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \le g$, is defined to hold iff: :$\forall s \in S: f \left({s}\right) \le g \left({s}\right)$ where $\le$ denotes the usual ordering on the extended real numbers $\overline{\R}$. Thence '''pointwise inequality''' of extended real-valued functions is an instance of an induced relation on mappings.
Definition:Increasing Sequence of Extended Real-Valued Functions
Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}, f_n: S \to \overline{\R}$ be a sequence of extended real-valued functions. Then $\left({f_n}\right)_{n \in \N}$ is said to be an '''increasing sequence (of extended real-valued functions)''' iff: :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \le f_n \left({s}\right)$ That is, iff $m \le n \implies f_m \le f_n$, where the second $\le$ denotes pointwise inequality.
Definition:Increasing Sequence of Real-Valued Functions
Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}, f_n: S \to \R$ be a sequence of real-valued functions. Then $\left({f_n}\right)_{n \in \N}$ is said to be an '''increasing sequence (of real-valued functions)''' : :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \le f_n \left({s}\right)$ That is,  $m \le n \implies f_m \le f_n$, where the second $\le$ denotes pointwise inequality.
Definition:Pointwise Supremum of Extended Real-Valued Functions
Let $S$ be a set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$ be an $I$-indexed collection of extended real-valued functions. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is taken in the extended real numbers $\overline \R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on extended real-valued functions.
Definition:Pointwise Supremum of Real-Valued Functions
Let $S$ be a set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \R$ be an $I$-indexed collection of real-valued functions. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is taken in the extended real numbers $\overline \R$. By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on real-valued functions. However, mind that this '''pointwise supremum''' need not be a real-valued function.
Definition:Pointwise Supremum
Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $\family {f_i}_{i \mathop \in I}, f_i: S \to T$ be an $I$-indexed collection of mappings. Suppose that for all $s \in S$, it holds that: :$\displaystyle \sup_{i \mathop \in I} \map {f_i} s \in T$ where the supremum is taken in $T$. Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to T$, is defined by: :$\displaystyle \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$ where the latter supremum is again taken in $T$. By assumption, this supremum is guaranteed to exist. Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation.
Definition:Ordered Commutative Semigroup
An '''ordered commutative semigroup''' is an ordered semigroup $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative semigroup.
Definition:Clearly
A statement is '''clearly''' true if its proof is apparent. Because of its subjective nature and the high likelihood of mistakes, this website does not endorse using such handwavery in any but the most trivial and boring of cases.
Definition:Divisor (Algebra)/Natural Numbers
Let $\N$ be the natural numbers. Let $n \in \N$ and $m \in \N_{>0}$. Then '''$m$ divides $n$''' is defined as: :$m \divides n \iff \exists p \in \N: m \times p = n$
Definition:Ordered Tuple/Defined by Sequence
Let $\sequence {a_k}_{k \mathop \in A}$ be a finite sequence of $n$ terms. Let $\sigma$ be a permutation of $A$. Then the '''ordered $n$-tuple defined by the sequence $\sequence {a_{\map \sigma k} }_{k \mathop \in A}$''' is the ordered $n$-tuple: :$\sequence {a_{\map \sigma {\map \tau j} } }_{1 \mathop \le j \mathop \le n}$ where $\tau$ is the unique isomorphism from the totally ordered set $\closedint 1 n$ onto the totally ordered set $A$.
Definition:Operation/N-Ary Operation
Let $S_1, S_2, \dots, S_n$ be sets. Let $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U$ be a mapping from the cartesian product $S_1 \times S_2 \times \ldots \times S_n$ to a universal set $\mathbb U$: That is, suppose that: :$\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \tuple {s_1, s_2, \ldots, s_n} \in S_1 \times S_2 \times \ldots \times S_n: \map \circ {s_1, s_2, \ldots, s_n} \in \mathbb U$ Then $\circ$ is an '''$n$-ary operation'''.
Definition:Operation/Operation on Set
An '''$n$-ary operation on a set $S$''' is an $n$-ary operation where: :the domain is the cartesian space $S^n$ :the codomain is $S$: :$\odot: S^n \to S: \forall \tuple {s_1, s_2, \ldots, s_n} \in S^n: \map \odot {s_1, s_2, \ldots, s_n} \in S$ That is: :an '''$n$-ary operation on $S$''' needs to be defined for ''all'' tuples in $S^n$ :the image of $\odot$ is itself in $S$.
Definition:Indexing Set/Index
An element of the domain $I$ of $x$ is called an '''index'''.
Definition:Indexing Set/Indexed Set
The image of $x$, that is, $x \sqbrk I$ or $\Img x$, is called an '''indexed set'''. That is, it is the '''set indexed by $I$'''
Definition:Indexing Set/Function
When used in this context, the mapping $x$ is referred to as an '''indexing function for $S$'''.
Definition:Indexing Set/Term
The image of $x$ at an index $i$ is referred to as a '''term''' of the (indexed) family, and is denoted $x_i$.
Definition:Indexing Set/Family of Distinct Elements
Let $x$ be an injection, that is: :$\forall \alpha, \beta \in I: \alpha \ne \beta \implies x_\alpha \ne x_\beta$ Then $\family {x_i} _{i \mathop \in I}$ is called a '''family of distinct elements of $S$'''.
Definition:Kernel (Measure Theory)
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\overline{\R}_{\ge0}$ be the set of positive extended real numbers. A '''kernel''' is a mapping $N: X \times \Sigma \to \overline{\R}_{\ge0}$ such that: :$(1):\quad \forall x \in X: N_x: \Sigma \to \overline{\R}_{\ge0}, E \mapsto N(x, E)$ is a measure :$(2):\quad \forall E \in \Sigma: N_E: X \to \overline{\R}_{\ge0}, x \mapsto N(x, E)$ is a positive $\Sigma$-measurable function {{expand|I found a more general approach on [http://www.physicsforums.com/showthread.php?t{{=}}352389 this website]}}
Definition:Kernel Transformation of Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. The '''transformation of $\mu$ by $N$''' is the mapping $\mu N: \Sigma \to \overline{\R}$ defined by: :$\displaystyle \forall E \in \Sigma: \mu N \left({E}\right) := \int N_E \left({x}\right) \, d\mu \left({x}\right)$ where $N_E \left({x}\right) = N \left({x, E}\right)$.
Definition:Kernel Transformation of Positive Measurable Function
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Let $f: X \to \overline{\R}$ be a positive measurable function. The '''transformation of $f$ by $N$''' is the mapping $N f: X \to \overline{\R}$ defined by: :$\forall x \in X: N f \left({x}\right) := \displaystyle \int f \, \mathrm dN_x$ where $N_x$ is the measure $E \mapsto N(x, E)$.
Definition:Integrable Function
=== Darboux === {{:Definition:Definite Integral/Darboux}} === Riemann === {{:Definition:Definite Integral/Riemann}} === Unbounded Above Positive Real Function === {{:Definition:Integrable Function/Unbounded Above}} === Unbounded Real Function === {{:Definition:Integrable Function/Unbounded}} === Measure Space === {{:Definition:Integrable Function/Measure Space}} === $p$-Integrable Function === {{:Definition:Integrable Function/p-Integrable}} === Lebesgue === {{:Definition:Integrable Function/Lebesgue}}
Definition:Internal Group Direct Product/General Definition
Let $\sequence {H_n} = \struct {H_1, \circ {\restriction_{H_1} } }, \ldots, \struct {H_n, \circ {\restriction_{H_n} } }$ be a (finite) sequence of subgroups of a group $\struct {G, \circ}$ where $\circ {\restriction_{H_1} }, \ldots, \circ {\restriction_{H_n} }$ are the restrictions of $\circ$ to $H_1, \ldots, H_n$ respectively. === Definition 1 === {{:Definition:Internal Group Direct Product/General Definition/Definition 1}} === Definition 2 === {{:Definition:Internal Group Direct Product/General Definition/Definition 2}}
Definition:Binomial Coefficient/Real Numbers
Let $r \in \R, k \in \Z$. Then $\dbinom r k$ is defined as: :$\dbinom r k = \begin{cases} \dfrac {r^{\underline k} } {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end{cases}$ where $r^{\underline k}$ denotes the falling factorial. That is, when $k \ge 0$: :$\displaystyle \binom r k = \frac {r \paren {r - 1} \cdots \paren {r - k + 1} } {k \paren {k - 1} \cdots 1} = \prod_{j \mathop = 1}^k \frac {r + 1 - j} j$ It can be seen that this agrees with the definition for integers when $r$ is an integer. For most applications the integer form is sufficient.
Definition:Integer/Formal Definition
Let $\struct {\N, +}$ be the commutative semigroup of natural numbers under addition. From Inverse Completion of Natural Numbers, we can create $\struct {\N', +'}$, an inverse completion of $\struct {\N, +}$. From Construction of Inverse Completion, this is done as follows: Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ From Cross-Relation is Congruence Relation, $\boxtimes$ is a congruence relation. Let $\struct {\N \times \N, \oplus}$ be the external direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$: :$\tuple {x_1, y_1} \oplus \tuple {x_2, y_2} = \tuple {x_1 + x_2, y_1 + y_2}$ Let the quotient structure defined by $\boxtimes$ be $\struct {\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}$ where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$. Let us use $\N'$ to denote the quotient set $\dfrac {\N \times \N} \boxtimes$. Let us use $+'$ to denote the operation $\oplus_\boxtimes$. Thus $\struct {\N', +'}$ is the Inverse Completion of Natural Numbers. As the Inverse Completion is Unique up to isomorphism, it follows that we can ''define'' the structure $\struct {\Z, +}$ which is isomorphic to $\struct {\N', +'}$. An element of $\N'$ is therefore an equivalence class of the congruence relation $\boxtimes$. So an element of $\Z$ is the isomorphic image of an element $\eqclass {\tuple {a, b} } \boxtimes$ of $\dfrac {\N \times \N} \boxtimes$. The set of elements $\Z$ is called '''the integers'''. === Natural Number Difference === {{:Construction of Inverse Completion/Natural Number Difference}}
Definition:Totally Ordered Commutative Semigroup
A '''totally ordered commutative semigroup''' is a totally ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative semigroup.
Definition:Multiplication/Integers
The multiplication operation in the domain of integers $\Z$ is written $\times$. Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers. That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$. $\boxtimes$ is the congruence relation defined on $\N \times \N$ by $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$. In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxtimes$, as suggested. As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus multiplication can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes. It follows that: :$\forall a, b, c, d \in \N: \eqclass {a, b} {} \times \eqclass {c, d} {} = \eqclass {a \times c + b \times d, a \times d + b \times c} {}$ or, more compactly, as $\eqclass {a c + b d, a d + b c} {}$. This can also be defined as: :$n \times m = +^n m = \underbrace {m + m + \cdots + m}_{\text{$n$ copies of $m$} }$ and the validity of this is proved in Index Laws for Monoids.
Definition:Ring (Abstract Algebra)/Product
The distributive operation $\circ$ in $\struct {R, *, \circ}$ is known as the '''(ring) product'''.
Definition:Ring (Abstract Algebra)/Ring Less Zero
It is convenient to have a symbol for $R \setminus \set 0$, that is, the set of all elements of the ring without the zero. Thus we usually use: :$R_{\ne 0} = R \setminus \set 0$
Definition:Non-Null Ring
A '''non-null ring''' is a ring with more than one element.
Definition:Integral of Integrable Function
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \overline \R$, $f \in \map {\LL^1} \mu$ be a $\mu$-integrable function. Then the '''$\mu$-integral of $f$''' is defined by: :$\displaystyle \int f \rd \mu := \int f^+ \rd \mu - \int f^- \rd \mu$ where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.
Definition:Space of Integrable Functions
Let $\struct {X, \Sigma, \mu}$ be a measure space. Then the '''space of $\mu$-integrable, real-valued functions''' $\map {\LL^1} \mu$ is the collection of all $\mu$-integrable real-valued functions: :$\map {\LL^1} \mu := \set {f: X \to \R: \text {$f$ is $\mu$-integrable} }$ Similarly, the '''space of $\mu$-integrable, extended real-valued functions''' $\map {\LL^1_{\overline \R} } \mu$ is the collection of all $\mu$-integrable extended real-valued functions: :$\map {\LL^1_{\overline \R} } \mu := \set {f: X \to \overline \R: \text {$f$ is $\mu$-integrable} }$
Definition:Integrable Function/Lebesgue
Let $\lambda^n$ be a Lebesgue measure on $\R^n$ for some $n > 0$. Let $f: \R^n \to \overline \R$ be an extended real-valued function. Then $f$ is said to be '''Lebesgue integrable'''  it is $\lambda^n$-integrable. Similarly, for all real numbers $p \ge 1$, $f$ is said to be '''Lebesgue $p$-integrable'''  it is $p$-integrable under $\lambda^n$.
Definition:Symmetric Matrix
Let $\mathbf A$ be a square matrix over a set $S$. $\mathbf A$ is '''symmetric''' : :$\mathbf A = \mathbf A^\intercal$ where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.
Definition:Vector Space Axioms
The '''vector space axioms''' consist of the abelian group axioms: {{begin-axiom}} {{axiom | n = \text V 0         | lc= Closure Axiom         | q = \forall \mathbf x, \mathbf y \in G         | m = \mathbf x +_G \mathbf y \in G }} {{axiom | n = \text V 1         | lc= Commutativity Axiom         | q = \forall \mathbf x, \mathbf y \in G         | m = \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x }} {{axiom | n = \text V 2         | lc= Associativity Axiom         | q = \forall \mathbf x, \mathbf y, \mathbf z \in G         | m = \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} }} {{axiom | n = \text V 3         | lc= Identity Axiom         | q = \exists \mathbf 0 \in G: \forall \mathbf x \in G         | m = \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 }} {{axiom | n = \text V 4         | lc= Inverse Axiom         | q = \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G         | m = \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 }} {{end-axiom}} together with the properties of a unitary module: {{begin-axiom}} {{axiom | n = \text V 5         | lc= Distributivity over Scalar Addition         | q = \forall \lambda, \mu \in K: \forall \mathbf x \in G         | m = \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x }} {{axiom | n = \text V 6         | lc= Distributivity over Vector Addition         | q = \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G         | m = \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y }} {{axiom | n = \text V 7         | lc= Associativity with Scalar Multiplication         | q = \forall \lambda, \mu \in K: \forall \mathbf x \in G         | m = \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x }} {{axiom | n = \text V 8         | lc= Identity for Scalar Multiplication         | q = \forall \mathbf x \in G         | m = 1_K \circ \mathbf x = \mathbf x }} {{end-axiom}}
Definition:Pairwise Disjoint/Set of Sets
A set of sets $\Bbb S$ is said to be '''pairwise disjoint''' : :$\forall X, Y \in \Bbb S: X \ne Y \implies X \cap Y = \O$ Here, $\cap$ denotes intersection, and $\O$ denotes the empty set. Hence we can say that the elements of $\Bbb S$ are '''pairwise disjoint'''.
Definition:Pairwise Disjoint/Family
An indexed family of sets $\family {S_i}_{i \mathop \in I}$ is said to be '''pairwise disjoint''' : :$\forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$ Hence the indexed sets $S_i$ themselves, where $i \in I$, are referred to as being '''pairwise disjoint'''.
Definition:Oscillation
Let $X$ be a set. Let $\left({Y, d}\right)$ be a metric space. Let $f: X \to Y$ be a mapping. === Oscillation on a Set === {{:Definition:Oscillation/Oscillation on Set}} === Oscillation at a Point === {{:Definition:Oscillation at Point}}
Definition:Null Ideal
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. The null ring $\struct {\set {0_R}, +, \circ}$ is called the '''null ideal''' of $R$.
Definition:Non-Null Ideal
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $J$ be an ideal of $R$ such that $J \ne \set {0_R}$. Then $J$ is a '''non-null ideal''' of $R$.
Definition:Only If
To say that: :''a statement $p$ is true '''only if''' $q$ is true'' is to say that: :''$p$ is a sufficient condition for $q$'': Thus: :$p \ \text{only if} \ q \dashv \vdash p \implies q$
Definition:If
To say that: :a statement $p$ is true '''if''' $q$ is true ...is to say that $p$ is a necessary condition for $q$: :$p \ \text{if} \ q \dashv \vdash q \implies p$
Definition:Internal Direct Sum of Rings/Direct Summand
In Conditions for Internal Ring Direct Sum it is proved that for this to be the case, then $S_1, S_2, \ldots, S_n$ must be ideals of $R$. Such ideals are known as '''direct summands''' of $R$.
Definition:Finite Integral Domain
A finite integral domain is an integral domain $\left({D, +, \circ}\right)$ such that $D$ is a finite set.  k6e73sa570prcefg8p2zj9yaswiy4ir
Definition:Standard Ordered Basis
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \in \closedint 1 n$, let $e_j = \tuple {0_R, 0_R, \ldots, 1_R, \cdots, 0_R}$ be the ordered $n$-tuple of elements of $R$ whose $j$th term is $1_R$ and all of whose other entries is $0_R$. Then the ordered $n$-tuple $\sequence {e_k}_{1 \mathop \le k \mathop \le n} = \tuple {e_1, e_2, \ldots, e_n}$ is called the '''standard ordered basis (of the $R$-module $R^n$)'''. === Vector Space === The concept of a '''standard ordered basis''' is often found in the context of vector spaces. {{:Definition:Standard Ordered Basis/Vector Space}}
Definition:Standard Basis
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\sequence {e_k}_{1 \mathop \le k \mathop \le n}$ be the standard ordered basis of the $R$-module $R^n$. The corresponding (unordered) set $\set {e_1, e_2, \ldots, e_n}$ is called the '''standard basis of $R^n$'''. === Vector Space === The concept of a '''standard basis''' is often found in the context of vector spaces. {{:Definition:Standard Basis/Vector Space}}
Definition:Unit Matrix
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ be the ring of order $n$ square matrices over $R$. Then the '''unit matrix (of order $n$)''' of $\struct {\map {\MM_R} n, +, \times}$ is defined as: :$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$ where $\delta_{i j}$ is the Kronecker delta for $\map {\MM_R} n$.
Definition:Positive/Integer
Informally, the '''positive integers''' are the set: :$\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$ As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus '''positive''' can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, '''positive''' can be defined directly as the relation specified as follows: :The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is '''positive'''  $b \le a$. The set of '''positive integers''' is denoted $\Z_{\ge 0}$. An element of $\Z$ can be specifically indicated as being '''positive''' by prepending a $+$ sign: :$+x := x \in \Z_{\ge 0}$. === Ordering on Integers === {{:Definition:Ordering on Integers}}
Definition:Positive/Number
The concept of positive can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ The Complex Numbers cannot be Ordered Compatibly with Ring Structure, so there is no such concept as a positive complex number. As for the natural numbers, they are all positive by dint of their being the non-negative integers. === Integers === {{:Definition:Positive/Integer}} === Rational Numbers === {{:Definition:Positive/Rational Number}} === Real Numbers === {{:Definition:Positive/Real Number}}
Definition:Ordering on Integers
=== Definition 1 === {{:Definition:Ordering on Integers/Definition 1}} === Definition 2 === {{:Definition:Ordering on Integers/Definition 2}}
Definition:Ordered Integral Domain/Trichotomy Law
The property: :$\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$ is known as the '''trichotomy law'''.
Definition:Positive/Rational Number
The '''positive rational numbers''' are the set defined as: :$\Q_{\ge 0} := \set {x \in \Q: x \ge 0}$ That is, all the rational numbers that are greater than or equal to zero.
Definition:Positive/Real Number
The '''positive real numbers''' are the set: :$\R_{\ge 0} = \set {x \in \R: x \ge 0}$ That is, all the real numbers that are greater than or equal to zero.
Definition:Strictly Positive/Number
The concept of strictly positive can be applied to the following sets of numbers: :$(1): \quad$ The integers $\Z$ :$(2): \quad$ The rational numbers $\Q$ :$(3): \quad$ The real numbers $\R$ === Integers === {{:Definition:Strictly Positive/Integer}} === Rational Numbers === {{:Definition:Strictly Positive/Rational Number}} === Real Numbers === {{:Definition:Strictly Positive/Real Number/Definition 1}}
Definition:Strictly Positive/Integer
The '''strictly positive integers''' are the set defined as: :$\Z_{> 0} := \set {x \in \Z: x > 0}$ That is, all the integers that are strictly greater than zero: :$\Z_{> 0} := \set {1, 2, 3, \ldots}$