hcju/proofwikips huggingface dataset:
Definition:Ordering/Size
An ordering can often be considered to be a comparison of the '''size''' of objects, perhaps in some intuitive sense. This is particularly applicable in the context of numbers. Thus the expression $A \preceq B$ can in such contexts be interpreted as:  : '''$A$ is smaller than $B$''' : '''$A$ is less than $B$''' and $B \preceq A$ can similarly be interpreted as:  : '''$A$ is larger than $B$''' : '''$A$ is greater than $B$''' In natural language, such terms are called '''[https://en.wikipedia.org/wiki/Comparative comparative adjectives]''', or just '''comparatives'''. Depending on the nature of the set being ordered, and depending on the nature of the ordering relation, this interpretation of an ordering as a comparison of size  may not be intellectually sustainable.
Definition:Strict Ordering/Notation
Symbols used to denote a general strict ordering are usually variants on $\prec$, $<$ and so on. On , to denote a general strict ordering it is recommended to use $\prec$. To denote the conventional strict ordering in the context of numbers, the symbol $<$ is to be used. The symbol $\subset$ is universally reserved for the (proper) subset relation. :$a \prec b$ can be read as: :'''$a$ (strictly) precedes $b$'''. Similarly: : $a \prec b$ can also be read as: : '''$b$ (strictly) succeeds $a$'''. If, for two elements $a, b \in S$, it is not the case that $a \prec b$, then the symbols $a \nprec b$ and $b \nsucc a$ can be used.
Definition:Inverse Relation/Strict Ordering Notation
To denote the inverse of an strict ordering, the conventional technique is to reverse the symbol. Thus: :$\succ$ denotes $\prec^{-1}$ and so: :$a \prec b \iff b \succ a$ Similarly for the standard symbol used to denote a strict ordering on numbers: :$>$ denotes $<^{-1}$ and so on.
Definition:Lattice (Ordered Set)
Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is a '''lattice''' : :for all $x, y \in S$, the subset $\left\{{x, y}\right\}$ admits both a supremum and an infimum.
Definition:Order Embedding/Definition 2
$\phi$ is an '''order embedding of $S$ into $T$'''  both of the following conditions hold: :$(1): \quad \phi$ is an injection :$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
Definition:Greatest/Ordered Set/Subset
Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is '''the greatest element of $T$''' : :$\forall y \in T: y \preceq \restriction_T x$ where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
Definition:Internal Direct Product/Decomposition
The set of algebraic substructures $\left({S_1, \circ {\restriction_{S_1}}}\right), \left({S_2, \circ {\restriction_{S_2}}}\right), \ldots, \left({S_n, \circ {\restriction_{S_n}}}\right)$ whose direct product is isomorphic with $\left({S, \circ}\right)$ is called a '''decomposition''' of $S$.
Definition:Internal Group Direct Product/Decomposition
The set of subgroups $\left({G_1, \circ {\restriction_{G_1}}}\right), \left({G_2, \circ {\restriction_{G_2}}}\right), \ldots, \left({G_n, \circ {\restriction_{G_n}}}\right)$ whose group direct product is isomorphic with $\left({G, \circ}\right)$ is called a '''decomposition''' of $G$.
Definition:Decomposable Group
Let $\left({G, \circ}\right)$ be a group. Then $\left({G, \circ}\right)$ is '''decomposable'''  there exists a decomposition of $\left({G, \circ}\right)$. That is,  $\left({G, \circ}\right)$ is the internal direct product of two (or more) proper subgroups of $G$. === Indecomposable === {{:Definition:Decomposable Group/Indecomposable}}
Definition:Decomposable Group/Indecomposable
$\left({G, \circ}\right)$ is '''indecomposable'''  it is not decomposable. That is,  there does not exist a decomposition of $\left({G, \circ}\right)$.
Definition:Lexicographic Order/Ordinals
The '''lexicographic order''' is a relation on ordered pairs of ordinals denoted $\operatorname{Le}$. $\operatorname{Le}$ is the set of all ordered pairs $\left({\left({\alpha, \beta}\right), \left({\gamma, \delta}\right)}\right)$ such that: :$(1): \quad$ Each $\alpha, \beta, \gamma, \delta$ is a member of the ordinal class :$(2): \quad$ $\alpha \in \gamma$ or $\alpha = \gamma \land \beta \in \delta$
Definition:Lexicographic Order/Tuples of Equal Length/Cartesian Space
Let $\left({S, \preceq}\right)$ be an ordered set. Let $n \in \N_{>0}$. Let $S^n$ be the cartesian $n$th power of $S$: :$S^n = \underbrace{S \times S \times \cdots \times S}_{n \text{ times} }$ The '''lexicographic order on $S^n$''' is the relation $\preccurlyeq$ defined on $S^n$ as: :$\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ : ::$\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec y_k}\right)$ :or: ::$\forall j: 1 \le j \le n: x_j = y_j$
Definition:Lexicographic Order/Tuples of Equal Length
Let $n \in \N_{>0}$. Let $\left({S_1, \preceq_1}\right), \left({S_2, \preceq_2}\right), \ldots, \left({S_n, \preceq_n}\right)$ be ordered sets. Let $\displaystyle S = \prod_{k \mathop = 1}^n S_k = S_1 \times S_2 \times \cdots \times S_n$ be the Cartesian product of $S_1$ to $S_n$. The '''lexicographic order on $S$''' is the relation $\preccurlyeq$ defined on $S$ as: :$\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ : ::$\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec_k y_k}\right)$ :or: ::$\forall j: 1 \le j \le n: x_j = y_j$
Definition:Lexicographic Order/General Definition
Let $\left({S, \preceq}\right)$ be an ordered set. For $n \in \N: n > 0$, we define $T_n$ as the set of all ordered $n$-tuples: :$\left({x_1, x_2, \ldots, x_n}\right)$ of elements $x_j \in S$. Let $\displaystyle T = \bigcup_{n \mathop \ge 1} T_n$. The '''lexicographic order on $T$''' is the relation $\preccurlyeq$ defined on $T$ as: :$\left({x_1, x_2, \ldots, x_m}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ : ::$\exists k: 1 \le k \le \min \left({m, n}\right): \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land x_k \prec y_k$ :or: ::$m \le n$ and $\forall j: 1 \le j \le m: x_j = y_j$.
Definition:Ordered Semigroup Axioms
An ordered semigroup is an algebraic structure $\struct {S, \circ, \preceq}$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = OS0         | lc= Closure         | q = \forall a, b \in S         | m = a \circ b \in S }} {{axiom | n = OS1         | lc= Associativity         | q = \forall a, b, c \in S         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = OS2         | lc= Compatibility of $\preceq$ with $\circ$         | q = \forall a, b, c \in S         | m = a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} }} {{axiom | lc= where $\preceq$ is an ordering         | m = a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} }} {{end-axiom}}
Definition:Ordered Group Axioms
An ordered group is an algebraic structure $\left({G, \circ, \preceq}\right)$ which satisfies the following properties: {{begin-axiom}} {{axiom | n = OG0         | lc= Closure         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = OG1         | lc= Associativity         | q = \forall a, b, c \in G         | m = a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c }} {{axiom | n = OG2         | lc= Identity         | q = \exists e \in G: \forall a \in G         | m = e \circ a = a = a \circ e }} {{axiom | n = OG3         | lc= Inverse         | q = \forall a \in G: \exists b \in G         | m = a \circ b = e = b \circ a }} {{axiom | n = OG4         | lc= Compatibility of $\preceq$ with $\circ$         | q = \forall a, b, c \in G         | m = a \preceq b \implies \left({a \circ c}\right) \preceq \left({b \circ c}\right) }} {{axiom | lc= where $\preceq$ is an ordering         | m = a \preceq b \implies \left({c \circ a}\right) \preceq \left({c \circ b}\right) }} {{end-axiom}}
Definition:Naturally Ordered Semigroup/Axioms
A '''naturally ordered semigroup''' is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying: {{begin-axiom}} {{axiom | n = \text {NO} 1         | lc= $S$ is well-ordered by $\preceq$         | q = \forall T \subseteq S         | m = T = \varnothing \lor \exists m \in T: \forall n \in T: m \preceq n }} {{axiom | n = \text {NO} 2         | lc= $\circ$ is cancellable in $S$         | q = \forall m, n, p \in S         | m = m \circ p = n \circ p \implies m = n }} {{axiom | m = p \circ m = p \circ n \implies m = n }} {{axiom | n = \text {NO} 3         | lc= Existence of product         | q = \forall m, n \in S         | m = m \preceq n \implies \exists p \in S: m \circ p = n }} {{axiom | n = \text {NO} 4         | lc= $S$ has at least two distinct elements         | q = \exists m, n \in S         | m = m \ne n }} {{end-axiom}}
Definition:Initial Segment of Natural Numbers/Zero-Based
Let $n \in \N$ be a natural number. The initial segment of the natural numbers determined by $n$: :$\set {0, 1, 2, \ldots, n - 1}$ is denoted $\N_{< n}$.
Definition:Initial Segment of Natural Numbers/One-Based
The initial segment of the non-zero natural numbers determined by $n$: :$\set {1, 2, 3, \ldots, n}$ is denoted $\N^*_{\le n}$.
Definition:Recursively Defined Mapping/Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $p \in S$. Let $S' = \set {x \in S: p \preceq x}$. Let $T$ be a set. Let $g: T \to T$ be a mapping. Let $f: S' \to T$ be the mapping defined as: :$\forall n \in S': \map f x = \begin{cases} a & : x = p \\ \map g {\map f n} & : x = n \circ 1 \end{cases}$ where $a \in T$. Then $f$ is said to be '''recursively defined''' on $S'$.
Definition:Recursively Defined Mapping
=== Natural Numbers === {{:Definition:Recursively Defined Mapping/Natural Numbers}}   === Peano Structure === {{:Definition:Recursively Defined Mapping/Peano Structure}} === Naturally Ordered Semigroup === {{:Definition:Recursively Defined Mapping/Naturally Ordered Semigroup}} === Minimal Infinite Successor Set === {{:Definition:Recursively Defined Mapping/Minimal Infinite Successor Set}}
Definition:Recursively Defined Mapping/Peano Structure
Let $\left({P, 0, s}\right)$ be a Peano structure. Let $T$ be a set. Let $g: T \to T$ be a mapping. Let  $f: P \to T$ be the mapping defined as: :$\forall x \in P: \map f x = \begin{cases}   a & : x = 0 \\   \map g {\map f n} & : x = \map s n \end{cases}$ where $a \in T$. Then $f$ is said to be '''recursively defined''' on $P$.
Definition:Recursively Defined Mapping/Minimal Infinite Successor Set
Let $\omega$ be the minimal infinite successor set. Let $T$ be a set. Let $a \in T$. Let $g: T \to T$ be a mapping. Let $f: \omega \to T$ be the mapping defined as: :$\forall x \in \omega: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = n^+ \end{cases}$ where $n^+$ is the successor set of $n$. Then $f$ is said to be '''recursively defined''' on $\omega$.
Definition:Recursively Defined Mapping/Natural Numbers
Let $p \in \N$ be a natural number. Let $S = \set {x \in \N: p \le x}$. Let $T$ be a set. Let $g: T \to T$ be a mapping. Let $f: S \to T$ be the mapping defined as: :$\forall x \in S: \map f x = \begin{cases} a & : x = p \\ \map g {\map f n} & : x = n + 1 \end{cases}$ for $a \in T$. Then $f$ is said to be '''recursively defined''' on $S$.
Definition:Power of Element/Magma with Identity
Let $\left({S, \circ}\right)$ be a magma with an identity element $e$. Let $a \in S$. Let the mapping $\circ^n a: \N \to S$ be recursively defined as: :$\forall n \in S: \circ^n a = \begin{cases} e & : n = 0 \\ \left({\circ^r a}\right) \circ a & : n = r + 1 \end{cases}$ The mapping $\circ^n a$ is known as the '''$n$th power of $a$ (under $\circ$)'''.
Definition:Power of Element/Magma
Let $\struct {S, \circ}$ be a magma which has no identity element. Let $a \in S$. Let the mapping $\circ^n a: \N_{>0} \to S$ be recursively defined as: :$\forall n \in \N_{>0}: \circ^n a = \begin{cases} a & : n = 1 \\ \paren {\circ^r a} \circ a & : n = r + 1 \end{cases}$ The mapping $\circ^n a$ is known as the '''$n$th power of $a$ (under $\circ$)'''.
Definition:Commutative Semiring
Let $\left({S, \circ, *}\right)$ be an additive semiring such that the distributor operation $*$ is a commutative operation. Then $\left({S, \circ}\right)$ is a '''commutative semiring'''.
Definition:Addition/Natural Numbers
{{:Definition:Addition in Peano Structure}}
Definition:Ordered Tuple/Term
Let $\sequence {a_k}_{k \mathop \in \N^*_n}$ be an ordered tuple. The ordered pair $\tuple {k, a_k}$ is called the '''$k$th term''' of the ordered tuple for each $k \in \N^*_n$.
Definition:Sequence/Notation
The notation for a sequence is as follows. If $f: A \to S$ is a sequence, then a symbol, for example "$a$", is chosen to represent elements of this sequence. Then for each $k \in A$, $f \left({k}\right)$ is denoted $a_k$, and $f$ itself is denoted $\left \langle {a_k} \right \rangle_{k \mathop \in A}$. Other types of brackets may be encountered, for example: :$\left({a_k}\right)_{k \mathop \in A}$ :$\left\{{a_k}\right\}_{k \mathop \in A}$ The latter is discouraged because of the implication that the order of the terms does not matter. Any expression can be used to denote the domain of $f$ in place of $k \in A$. For example: :$\left \langle {a_k} \right \rangle_{k \mathop \ge n}$ :$\left \langle {a_k} \right \rangle_{p \mathop \le k \mathop \le q}$ The sequence itself may be defined by a simple formula, and so for example: :$\left \langle {k^3} \right \rangle_{2 \mathop \le k \mathop \le 6}$ is the same as: :$\left \langle {a_k} \right \rangle_{2 \mathop \le k \mathop \le 6}$ where $a_k = k^3$ for all $k \in \left\{{2, 3, \ldots, 6}\right\}$. The set $A$ is usually taken to be the set of natural numbers $\N = \left\{{0,1, 2, 3, \ldots}\right\}$ or a subset. In particular, for a finite sequence, $A$ is usually $\left\{{0, 1, 2, \ldots, n-1}\right\}$ or $\left\{{1, 2, 3, \ldots, n}\right\}$. If this is the case, then it is usual to write $\left \langle {a_k} \right \rangle_{k \mathop \in A}$ as $\left \langle {a_k} \right \rangle$ or even as $\left \langle {a} \right \rangle$ if brevity and simplicity improve clarity. A finite sequence of length $n$ can be denoted: :$\left({a_1, a_2, \ldots, a_n}\right)$ and by this notational convention the brackets are always round.
Definition:Sequence/Equality
Let $f$ and $g$ be two sequences on the same set $A$: : $f = \left\langle{a_k}\right\rangle_{k \mathop \in A}$ : $g = \left\langle{b_k}\right\rangle_{k \mathop \in B}$ Then $f = g$ : : $A = B$ : $\forall i \in A: a_i = b_i$
Definition:Empty Set/Existence
Some authors have problems with the existence (or not) of the empty set: * {{BookReference|Sets and Groups|1965|J.A. Green}}: $\S 1.3$: :: ''If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional ''empty set'', denoted $\O$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\O$ the status of a set.'' * {{BookReference|The Theory of Groups|1968|Ian D. Macdonald}}: Appendix: :: ''The best attitude towards the empty set $\O$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \O$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\O$ is a subset of every set, for elements of $\O$ are supposed to possess every property.'' * {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts :: ''Now some students are bothered with the notion of an "empty set". "How", they say, "can you have a set with nothing in it?" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.'' Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero. In any case, its convenience cannot be doubted: * {{BookReference|Lectures in Abstract Algebra|1951|Nathan Jacobson|volume = I|subtitle = Basic Concepts}}: Introduction $\S 1$: Operations on Sets: :: ''One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.'' * {{BookReference|Modern Algebra|1965|Seth Warner}}: $\S 1$: :: ''One practical advantage in admitting $\O$ as a set is that we may wish to talk about a set without knowing ''a priori'' whether it has any members.'' * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}}: $\S 1$: :: ''The courtesy of regarding this as a set has several advantages ... In allowing $\O$ the status of a set, we gain the advantage of being able to talk about a set without knowing at the outset whether or not it has any elements.'' Other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite.
Definition:Topologically Distinguishable/Indistinguishable
The two points $x$ and $y$ are '''topologically indistinguishable'''  they are not '''topologically distinguishable'''. That is: :$\forall U \in \tau: x \in U \iff y \in U$
Definition:Strict Upper Closure/Element
Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $a \in S$. The '''strict upper closure of $a$ (in $S$)''' is defined as: :$a^\succ := \left\{{b \in S: a \preccurlyeq b \land a \ne b}\right\}$ or: :$a^\succ := \left\{{b \in S: a \prec b}\right\}$ That is, $a^\succ$ is the set of all elements of $S$ that strictly succeed $a$.
Definition:Strict Upper Closure/Set
Let $\left({S, \preceq}\right)$ be an ordered set or preordered set. Let $T \subseteq S$. The '''strict upper closure of $T$ (in $S$)''' is defined as: :$T^\succ := \bigcup \left\{{t^\succ: t \in T}\right\}$ where $t^\succ$ denotes the strict upper closure of $t$ in $S$. That is: :$T^\succ := \left\{ {u \in S: \exists t \in T: t \prec u}\right\}$
Definition:Strict Lower Closure/Element
Let $\left({S, \preccurlyeq}\right)$ be an ordered set. Let $a \in S$. The '''strict lower closure of $a$ (in $S$)''' is defined as: :$a^\prec := \left\{{b \in S: b \preccurlyeq a \land a \ne b}\right\}$ or: :$a^\prec := \left\{{b \in S: b \prec a}\right\}$ That is, $a^\prec$ is the set of all elements of $S$ that strictly precede $a$.
Definition:Strict Lower Closure/Set
Let $\left({S, \preceq}\right)$ be an ordered set or a preordered set. Let $T \subseteq S$. The '''strict lower closure of $T$ (in $S$)''' is defined as: :$T^\prec := \displaystyle \bigcup \left\{{t^\prec: t \in T}\right\}$ where $t^\prec$ denotes the strict lower closure of $t$ in $S$. That is: :$T^\prec := \left\{ {u \in S: \exists t \in T: u \prec t}\right\}$
Definition:Real Interval/Notation/Wirth
The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by {{AuthorRef|Niklaus Emil Wirth}}: {{begin-eqn}} {{eqn | l = \openint a b       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open Real Interval }} {{eqn | l = \hointr a b       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-Open (to the right) Real Interval }} {{eqn | l = \hointl a b       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-Open (to the left) Real Interval }} {{eqn | l = \closedint a b       | o = :=       | r = \set {x \in \R: a \le x \le b}       | c = Closed Real Interval }} {{end-eqn}} The term '''Wirth interval notation''' has consequently been coined by .
Definition:Real Interval/Notation/Unbounded Intervals
Some authors (sensibly, perhaps) prefer not to use the $\infty$ symbol and instead use $\to$ and $\gets$ for $+\infty$ and $-\infty$ repectively. In Wirth interval notation, such intervals are written as follows: {{begin-eqn}} {{eqn | l = \hointr a \to       | o = :=       | r = \set {x \in \R: a \le x} }} {{eqn | l = \hointl \gets a       | o = :=       | r = \set {x \in \R: x \le a} }} {{eqn | l = \openint a \to       | o = :=       | r = \set {x \in \R: a < x} }} {{eqn | l = \openint \gets a       | o = :=       | r = \set {x \in \R: x < a} }} {{eqn | l = \openint \gets \to       | o = :=       | r = \set {x \in \R} = \R }} {{end-eqn}}
Definition:Strictly Totally Ordered Set
A '''strictly totally ordered set''' is a relational structure $\left({S, \prec}\right)$ such that the relation $\prec$ is a strict total ordering.
Definition:Strictly Partially Ordered Set
A '''strictly partially ordered set''' is a relational structure $\left({S, \prec}\right)$ such that the relation $\prec$ is an strict partial ordering.
Definition:Addition/Peano Structure
Let $\struct {P, 0, s}$ be a Peano structure. The binary operation $+$ is defined on $P$ as follows: :$\forall m, n \in P: \begin{cases} m + 0 & = m \\ m + \map s n & = \map s {m + n} \end{cases}$ This operation is called '''addition'''.
Definition:Real Number/Axioms
The properties of the field of real numbers $\struct {R, +, \times, \le}$ are as follows: {{begin-axiom}} {{axiom | n = \R \text A 0         | lc= Closure under addition         | q = \forall x, y \in \R         | m = x + y \in \R }} {{axiom | n = \R \text A 1         | lc= Associativity of addition         | q = \forall x, y, z \in \R         | m = \paren {x + y} + z = x + \paren {y + z} }} {{axiom | n = \R \text A 2         | lc= Commutativity of addition         | q = \forall x, y \in \R         | m = x + y = y + x }} {{axiom | n = \R \text A 3         | lc= Identity element for addition         | q = \exists 0 \in \R: \forall x \in \R         | m = x + 0 = x = 0 + x }} {{axiom | n = \R \text A 4         | lc= Inverse elements for addition         | q = \forall x: \exists \paren {-x} \in \R         | m = x + \paren {-x} = 0 = \paren {-x} + x }} {{axiom | n = \R \text M 0         | lc= Closure under multiplication         | q = \forall x, y \in \R         | m = x \times y \in \R }} {{axiom | n = \R \text M 1         | lc= Associativity of multiplication         | q = \forall x, y, z \in \R         | m = \paren {x \times y} \times z = x \times \paren {y \times z} }} {{axiom | n = \R \text M 2         | lc= Commutativity of multiplication         | q = \forall x, y \in \R         | m = x \times y = y \times x }} {{axiom | n = \R \text M 3         | lc= Identity element for multiplication         | q = \exists 1 \in \R, 1 \ne 0: \forall x \in \R         | m = x \times 1 = x = 1 \times x }} {{axiom | n = \R \text M 4         | lc= Inverse elements for multiplication         | q = \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}         | m = x \times \frac 1 x = 1 = \frac 1 x \times x }} {{axiom | n = \R \text D         | lc= Multiplication is distributive over addition         | q = \forall x, y, z \in \R         | m = x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} }} {{axiom | n = \R \text O 1         | lc= Usual ordering is compatible with addition         | q = \forall x, y, z \in \R         | m = x > y \implies x + z > y + z }} {{axiom | n = \R \text O 2         | lc= Usual ordering is compatible with multiplication         | q = \forall x, y, z \in \R         | m = x > y, z > 0 \implies x \times z > y \times z }} {{axiom | n = \R \text O 3         | lc= $\struct {R, +, \times, \le}$ is Dedekind complete         | q =          | m =  }} {{end-axiom}} These are called the '''real number axioms'''.
Definition:Indexing Set/Family
The image $\Img x$, consisting of the terms $\family {x_i}_{i \mathop \in I}$, along with the indexing function $x$ itself, is called a '''family of elements of $S$ indexed by $I$'''.