Definition:Hasse Diagram/Examples

These are examples of Hasse diagrams : :350px $\qquad$ 350px The diagram on the left illustrates the "Divisor" ordering on the set $S = \left\{{1, 2, 3, 4, 6, 8, 12, 24}\right\}$ where $S$ is the set of all elements of $\N_{>0}$ which divide $24$. The diagram on the right illustrates the "Subset" relation on the power set $\mathcal P \left({S}\right)$ where $S = \left\{{1, 2, 3}\right\}$.


Definition:Smallest/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 smallest element of $T$''' {{iff}}: :$\forall y \in T: x \preceq \restriction_T y$ where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.


Definition:Inverse Relation/Ordering Notation

To denote the inverse of an ordering, the conventional technique is to reverse the symbol. Thus: :$\succeq$ denotes $\preceq^{-1}$ :$\succcurlyeq$ denotes $\preccurlyeq^{-1}$ :$\curlyeqsucc$ denotes $\curlyeqprec^{-1}$ and so: :$a \preceq b \iff b \succeq a$ :$a \preccurlyeq b \iff b \succcurlyeq a$ :$a \curlyeqprec b \iff b \curlyeqsucc a$ Similarly for the standard symbols used to denote an ordering on numbers: :$\ge$ denotes $\le^{-1}$ :$\geqslant$ denotes $\leqslant^{-1}$ :$\eqslantgtr$ denotes $\eqslantless^{-1}$ and so on.


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 {{ProofWiki}}, 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''' {{iff}}: :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 $T$''' {{iff}} 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 $T$''' {{iff}}: :$\forall y \in T: y \preceq \restriction_T x$ 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''' {{iff}} there exists a decomposition of $\left({G, \circ}\right)$. That is, {{iff}} $\left({G, \circ}\right)$ 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''' {{iff}} it is not decomposable. That is, {{iff}} 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: :$\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ {{iff}}: ::$\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec y_k}\right)$ :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: :$\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ {{iff}}: ::$\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec_k y_k}\right)$ :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: :$\left({x_1, x_2, \ldots, x_m}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ {{iff}}: ::$\exists k: 1 \le k \le \min \left({m, n}\right): \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land x_k \prec y_k$ :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'''. Category:Definitions/Ring Theory 4b5tf6wyezwhoe4hpp66jxq652a2u2e


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 $f = g$ {{iff}}: : $A = B$ : $\forall i \in A: a_i = b_i$


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''' {{iff}} 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 {{ProofWiki}}.


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/Note on Terminology

It is a common approach to blur the distinction between an indexing function $x: I \to S$ and the indexed family $\family {x_i}_{i \mathop \in I}$ itself, and refer to the mapping as the indexed family. This approach is in accordance with the definition of a mapping as a relation defined as an as ordered triple $\tuple {I, S, x}$, where the mapping is understood as being ''defined'' to include its domain and codomain. However, on {{ProofWiki}} the approach is taken to separate the concepts carefully such that an indexed family is defined as: :the set of terms of the indexed set together with: :the indexing function itself denoting the combination as $\family {x_i}_{i \mathop \in I}$. The various approaches in the literature can be exemplified as follows. :''There are occasions when the range of a function is deemed to be more important than the function itself. When that is the case, both the terminology and the notation undergo radical alterations. Suppose, for instance, that $x$ is a function from a set $I$ to a set $X$. ... An element of the domain $I$ is called an '''index''', $I$ is called the '''index set''', the range of the function is called an '''indexed set''', the function itself is called a '''family''', and the value of the function $x$ at an index $i$, called a '''term''' of the family, is denoted by $x_i$. (This terminology is not absolutely established, but it is one of the standard choices among related slight variants...) An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family $\set {x_i}$ in $X$, or of a family $\set {x_i}$ of whatever the elements of $X$ may be; when necessary, the index set $I$ is indicated by some such parenthetical expression as $\paren {i \in I}$. Thus, for instance, the phrase "a family $\set {A_i}$ of subsets of $X$" is usually understood to refer to a function $A$, from some set $I$ of indices, into $\powerset X$.'' :::{{BookReference|Naive Set Theory|1960|Paul R. Halmos}}: $\S 9$: Families :''Occasionally, the special notation for sequences is also employed for functions that are not sequences. If $f$ is a function from $A$ into $E$, some letter or symbol is chosen, say "$x$", and $\map f \alpha$ is denoted by $x_\alpha$ and $f$ itself by $\paren {x_\alpha}_{\alpha \mathop \in A}$. When this notation is used, the domain $A$ of $f$ is called the set of '''indices''' of $\paren {x_\alpha}_{\alpha \mathop \in A}$, and $\paren {x_\alpha}_{\alpha \mathop \in A}$ is called a '''family of elements of $E$ indexed by $A$''' instead of a function from $A$ into $E$.'' :::{{BookReference|Modern Algebra|1965|Seth Warner}}: $\S 18$: The Natural Numbers :''Let $I$ and $E$ be sets and let $f: I \to E$ be a mapping, described by $i \mapsto \map f i$ for each $i \in I$. We often find it convenient to write $x_i$ instead of $\map f i$ and write the mapping as $\paren {x_i}_{i \mathop \in I}$ which we shall call a '''family of elements of $E$ indexed by $I$'''. By abuse of language we refer to the $x_i$ as the '''elements of the family'''.'' :''...'' :''As we have already mentioned, many authors identify a mapping with its graph, thereby identifying the family $\paren {x_i}_{i \mathop \in I}$ with the set $\set {\tuple {i, x_i}; i \in I}$. In the case where the elements of the family are all distinct, some authors go even further and identify the mapping $\paren {x_i}_{i \mathop \in I}$ with its image $\set {x_i; i \in I}$.'' :::{{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth}}: $\S 6$. Indexed families; partitions; equivalence relations Some authors are specific about the types of objects to which this construction is applied: :''Let $\mathcal A$ be a nonempty collection of sets. An '''indexing function''' for $\mathcal A$ is a surjective function $f$ from some set $J$, called the '''index set''', to $\mathcal A$. The collection $\mathcal A$, together with the indexing function $f$, is called an '''indexed family of sets'''. Given $\alpha \in J$, we shall denote the set $\map f \alpha$ by the symbol $\mathcal A_\alpha$. And we shall denote the indexed family itself by the symbol $\set {\mathcal A_\alpha}_{\alpha \mathop \in J}$, which is read as "the family of all $\mathcal A_\alpha$, as $\alpha$ ranges over $J$."'' :::{{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}: $\S 5$: Cartesian Products


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 $I$'''.


Definition:Indexing Set/Notation

The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations: :$\family {x_i}_{i \mathop \in I}$ :$\paren {x_i}_{i \mathop \in I}$ :$\set {x_i}_{i \mathop \in I}$ There is little consistency in the literature, but $\paren {x_i}_{i \mathop \in I}$ is perhaps most common. The preferred notation on {{ProofWiki}} is $\family {x_i}_{i \mathop \in I}$. The subscripted $i \in I$ is often left out, if it is obvious in the particular context. Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$. As $x$ is actually a mapping, one would expect the conventional notation $\map x i$. However, this is generally not used, and $x_i$ is used instead.


Definition:Indexing Set/Family of Subsets

Let $S$ be a set. Let $I$ be an indexing set. For each $i \in I$, let $S_i$ be a corresponding subset of $S$. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is referred to as an '''indexed family of subsets (of $S$ by $I$)'''.


Definition:Cartesian Product/Cartesian Space/Real Cartesian Space/Countable

The countable cartesian product defined as: :$\displaystyle \R^\omega := \R \times \R \times \cdots = \prod_\N \R$ is called the '''countable-dimensional real cartesian space'''. Thus, $\R^\omega$ can be defined as the set of all real sequences: :$\R^\omega = \left\{{\left({x_1, x_2, \ldots}\right): x_1, x_2, \ldots \in \R}\right\}$


Definition:Composite Defined by Permutation

Let $\oplus$ be an $n$-ary operation on a set $S$. Let $\left\langle{a_k}\right\rangle_{k \mathop \in A}$ be a sequence of $n$ terms of $S$. Let $\sigma: A \to A$ be a permutation of $A$. Then the '''composite of the ordered $n$-tuple defined by the sequence $\left\langle{a_{\sigma \left({k}\right)}}\right\rangle_{k \mathop \in A}$''' is defined as: {{begin-eqn}} {{eqn | l = \bigoplus_{k \mathop \in A} a_{\sigma \left({k}\right)}       | r = \begin{cases} a_{\sigma \left({1}\right)} & : n = 1 \\ \oplus \left({a_{\sigma \left({1}\right)}, a_{\sigma \left({2}\right)}, \ldots, a_{\sigma \left({m}\right)} }\right) \oplus a_{\sigma \left({n + 1}\right)} & : n = m + 1 \end{cases}       | c =  }} {{eqn | r = a_{\sigma \left({1}\right)} \oplus a_{\sigma \left({2}\right)} \oplus \cdots  \oplus a_{\sigma \left({n}\right)}       | c =  }} {{end-eqn}}


Definition:Power of Element/Semigroup

Let $\struct {S, \circ}$ be a semigroup which has no identity element. Let $a \in S$. For $n \in \N_{>0}$, the $n$th power of $a$ (under $\circ$) is defined as: :$\circ^n a = \begin{cases} a & : n = 1 \\ \paren {\circ^m a} \circ a & : n = m + 1 \end{cases}$ That is: :$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a}$ which from the General Associativity Theorem is unambiguous.


Definition:Incircle/Incenter

The center of an incircle of a polygon is called an '''incenter of the polygon'''.


Definition:Incircle/Inradius

A radius of an incircle of a polygon is called an '''inradius of the polygon'''. In the above diagram, $OG$ is an inradius of the polygon $ABCDEF$.


Definition:Independent Subgroups/Definition 2

The subgroups $H_1, H_2, \ldots, H_n$ are '''independent''' {{iff}}: :$\displaystyle \forall k \in \set {2, 3, \ldots, n}: \paren {\prod_{j \mathop = 1}^{k - 1} H_j} \cap H_k = \set e$


Definition:Independent Subgroups/Definition 1

The subgroups $H_1, H_2, \ldots, H_n$ are '''independent''' {{iff}}: :$\displaystyle \prod_{k \mathop = 1}^n h_k = e \iff \forall k \in \set {1, 2, \ldots, n}: h_k = e$ where $h_k \in H_k$ for all $k \in \set {1, 2, \ldots, n}$.


Definition:Factorial/Historical Note

The symbol $!$ used on {{ProofWiki}} for the factorial, which is now universal, was introduced by {{AuthorRef|Christian Kramp}} in his $1808$ work {{BookLink|Élémens d'arithmétique universelle|C. Kramp}}. Before that, various symbols were used whose existence is now of less importance. Notations for $n!$ in history include the following: :$\sqbrk n$ as used by {{AuthorRef|Leonhard Paul Euler|Euler}} :$\mathop{\Pi} n$ as used by {{AuthorRef|Carl Friedrich Gauss|Gauss}} :$\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy. {{AuthorRef|Augustus De Morgan}} declared his reservations about {{AuthorRef|Christian Kramp|Kramp}}'s notation thus: :''Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration  that $2$, $3$, $4$, etc., should be found in mathematical results.'' ::::-- {{BookReference|A History of Mathematical Notations|1929|Florian Cajori|volume = 2}} The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead.


Definition:Factorial/Multiindices

Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \geq 0$. Then we define: :$\displaystyle\alpha! = \prod_{j \mathop \in J} \alpha_j!$ where the factorial on the right is a factorial of natural numbers. Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.


Definition:Empty Topological Space

Let $\varnothing$ denote the empty set. The topological space $\left({\varnothing, \left\{ {\varnothing}\right\} }\right)$ is called the '''empty topological space'''.


Definition:Topology/Definition 1

Let $S$ be a set. A '''topology on $S$''' is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the open set axioms: {{:Definition:Open Set Axioms}} If $\tau$ is a '''topology''' on $S$, then $\struct {S, \tau}$ is called a topological space. The elements of $\tau$ are called the open sets of $\struct {S, \tau}$.


Definition:Topology/Definition 2

Let $S$ be a set. A '''topology on $S$''' is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the following axioms: {{begin-axiom}} {{axiom | n = \text O 1'         | t = The union of an arbitrary subset of $\tau$ is an element of $\tau$. }} {{axiom | n = \text O 2'         | t = The intersection of any finite subset of $\tau$ is an element of $\tau$. }} {{end-axiom}}


Definition:Neighborhood (Topology)/Neighborhood defined as Open

Some authorities define a neighborhood of a set $A$ as what {{ProofWiki}} defines as an open neighborhood: :$N_A$ is a '''neighborhood of $A$''' {{iff}} $N_A$ is an open set of $T$ which itself contains $A$. That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$. However, this treatment is less common, and considered by many to be old-fashioned. When the term '''neighborhood''' is used on this site, it is assumed to be not necessarily open unless so specified.


Definition:Basis (Topology)/Analytic Basis/Definition 1

Let $\struct {S, \tau}$ be a topological space. An '''analytic basis for $\tau$''' is a subset $\BB \subseteq \tau$ such that: :$\displaystyle \forall U \in \tau: \exists \AA \subseteq \BB: U = \bigcup \AA$ That is, such that for all $U \in \tau$, $U$ is a union of sets from $\BB$.


Definition:Basis (Topology)/Analytic Basis/Definition 2

Let $\left({S, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$. Then $\mathcal B$ is an analytic basis for $\tau$ {{iff}}: :$\forall U \in \tau: \forall x \in U: \exists V \in \mathcal B: x \in V \subseteq U$


Definition:Isolated Point (Topology)/Subset/Definition 1

$x \in H$ is an '''isolated point of $H$''' {{iff}}: :$\exists U \in \tau: U \cap H  = \set x$ That is, {{iff}} there exists an open set of $T$ containing no points of $H$ other than $x$.


Definition:Isolated Point (Topology)/Subset/Definition 2

$x \in H$ is an '''isolated point of $H$''' {{iff}} $x$ is not a limit point of $H$. That is, {{iff}} $x$ is not in the derived set of $H$.


Definition:Addition in Minimal Infinite Successor Set

Let $\omega$ be the minimal infinite successor set. The binary operation $+$ is defined on $\omega$ as follows: :$\forall m,n \in \omega: \begin{cases} m + 0 &= m \\ m + n^+ &= \left({m + n}\right)^+\end{cases}$ where $m^+$ is the successor set of $m$. This operation is called '''addition'''.


Definition:Addition for Natural Numbers in Real Numbers

Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Then the restriction of $+$ to $\N$ is called '''addition'''.


Definition:Closed Set/Topology/Definition 1

'''$H$ is closed (in $T$)''' {{iff}} its complement $S \setminus H$ is open in $T$.  That is, $H$ is '''closed''' {{iff}} $\paren {S \setminus H} \in \tau$. That is, {{iff}} $S \setminus H$ is an element of the topology of $T$.


Definition:Closed Set/Topology/Definition 2

'''$H$ is closed (in $T$)''' {{iff}} every limit point of $H$ is also a point of $H$. That is, by the definition of the derived set: :'''$H$ is closed (in $T$)''' {{iff}} $H' \subseteq H$ where $H'$ denotes the derived set of $H$.


Definition:Scaled Euclidean Metric

Let $\R_{>0}$ be the set of strictly positive real numbers. Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as: :$\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$ Then $\delta$ is the '''scaled Euclidean metric''' on $\R_{>0}$.


Definition:Alexandroff Extension

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open {{iff}}: :$U$ is an open set of $T$ or :$U$ is the complement in $T^*$ of a closed and compact subset of $T$. This topology is called the '''Alexandroff extension''' on $S$.


Definition:Centroid of Triangle

Let $\triangle ABC$ be a triangle. The '''centroid''' of $\triangle ABC$ is where its three medians meet. :500px


Definition:One-Sided Limit of Real Function

A '''one-sided limit of real function''' is a right-hand limit of real function or a left-hand limit of real function: === Limit from Right === {{:Definition:Limit of Real Function/Right}} === Limit from Left === {{:Definition:Limit of Real Function/Left}} Category:Definitions/Real Analysis Category:Definitions/Limits k8m8gaomjoc9y8kcpjw3dzujindqdpf


Definition:Functor/Contravariant/Definition 1

Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''contravariant functor''' $F : \mathbf C \to \mathbf D$ consists of: * An '''object functor''' $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$. * An '''arrow functor''' $F_1$ that assigns to each arrow $f : X \to Y$ of $\mathbf C$ an arrow $Ff : FY \to FX$ of $\mathbf D$. These '''functors''' must satisfy, for any morphisms $X \stackrel f \longrightarrow Y \stackrel g \longrightarrow Z$ in $\mathbf C$: :$\map F {g \circ f} = F f \circ F g$ and: :$\map F {\operatorname {id}_X} = \operatorname {id}_{F X}$ where: :$\operatorname {id}_W$ denotes the identity arrow on an object $W$ and: :$\circ$ is the composition of morphisms.


Definition:Functor/Contravariant/Definition 2

Let $\mathbf C$ and $\mathbf D$ be metacategories. A '''contravariant functor''' $F : \mathbf C \to \mathbf D$ is a covariant functor: :$F: \mathbf C^{\text{op}} \to \mathbf D$ where $\mathbf C^{\text{op}}$ is the dual category of $\mathbf C$.


Definition:Normal Extension/Definition 1

Let $L / K$ be a field extension. Then $L / K$ is a '''normal extension''' {{iff}}: : for every irreducible polynomial $f \in K \left[{x}\right]$ with at least one root in $L$, $f$ splits completely in $L$.


Definition:Normal Extension/Definition 2

Let $L / K$ be a field extension. Let $\overline K$ be the algebraic closure of $K$. Let $\operatorname{Gal} \left({L / K}\right)$ denote the set of embeddings of $L$ in $\overline K$ which fix $K$ pointwise. Then $L/K$ is a '''normal extension''' {{iff}}: : $\sigma \left({L}\right) = L$ for each $\sigma \in \operatorname{Gal} \left({L / K}\right)$.


Definition:Decomposable Element

Let $\left({D, +, \circ}\right)$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there exist a complete factorization of $x$ in $D$. Then $x$ is '''decomposable'''. Category:Definitions/Factorization 0zsw4x5af6xzpnef1uvc2tit86x6fwu


Definition:Zariski Topology/Affine Space

Let $k$ be a field. Let $\mathbb A^n \left({k}\right) = k^n$ denote the standard affine space of dimension $n$ over $k$. The '''Zariski topology''' on $\mathbb A^n \left({k}\right)$ is the topology on the direct product $k^n$ whose closed sets are the affine algebraic sets in $\mathbb A^n \left({k}\right)$.


Definition:Zariski Topology/Spectrum of Ring

Let $A$ be a commutative ring with unity. Let $\operatorname{Spec} \left({A}\right)$ be the prime spectrum of $A$. The '''Zariski topology''' on $\operatorname{Spec} A$ is the topology with closed sets the vanishing sets $V \left({S}\right)$ for $S \subseteq A$.


Definition:Logarithm/Base

Let $\log_a$ denote the logarithm function on whatever domain: $\R$ or $\C$. The constant $a$ is known as the '''base''' of the logarithm.


Definition:Tangent Vector/Definition 1

A '''tangent vector $X_m$ on $M$ at $m$''' is a linear transformation: :$X_m: C^\infty \left({V, \R}\right) \to \R$ which satisfies the Leibniz law: :$\displaystyle X_m \left({f g}\right) = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)$


Definition:Tangent Vector/Definition 2

Let $I$ be an open real interval with $0 \in I$.  Let $\gamma: I \to M$ be a smooth curve with $\gamma \left({0}\right) = m$.  Then a '''tangent vector''' $X_m$ at a point $m \in M$ is a mapping  :$X_m: C^\infty \left({V, \R}\right) \to \R$ defined by: :$X_m \left({f} \right) := \dfrac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, f \circ \gamma \left({\tau}\right)$ for all $f \in C^\infty \left({V, \R}\right)$.


Definition:Inverse Laplace Transform

=== Definition 1 === {{:Definition:Inverse Laplace Transform/Definition 1}} === Definition 2 === {{:Definition:Inverse Laplace Transform/Definition 2}}


Definition:Inverse Laplace Transform/Definition 2

Let $\map f s: S \to \R$ be a complex function, where $S \subset \R$. {{mistake|How can it be a complex function when both its domain and codomain are wholly real?}} The '''inverse Laplace transform''' of $f$, denoted $\map F t: \R \to S$, is defined as: :$\map F t = \dfrac 1 {2 \pi i} \PV_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s$ where: :$\PV$ is the Cauchy principal value of the integral :$c$ is any real constant such that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.


Definition:Inverse Laplace Transform/Definition 1

Let $f \left({s}\right) : S \to \R$ be a function, where $S \subset \R$. Let $F$ be another function such that $F$ is the Laplace transform of $f$. Then, $f$ is the '''inverse Laplace transform''' of $F$.


Definition:Topological Surface

A '''surface''', in the context of topology, is a smooth topological manifold whose dimension is $2$. More briefly, a '''surface''' is a smooth $2$-manifold. Category:Definitions/Topology Category:Definitions/Algebraic Topology d03a75ochzeskx4zmadmgjvolmjlpax


Definition:Taylor Polynomial

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and $n + 1$ times differentiable on the open interval $\openint a b$. Let $\xi \in \openint a b$. The polynomial $T_n$ defined as:  :$\displaystyle \map {T_n} x = \sum_{i \mathop = 0}^i \frac {\paren {x - \xi}^i} {i!} \map {f^{\paren i} } \xi$ is known as the '''Taylor polynomial of degree $n$ for $f$ about $\xi$'''. That is, a '''Taylor polynomial''' is a Taylor series taken for $n$ initial terms.


Definition:Bessel Function

The '''Bessel functions''' are solutions to Bessel's equation: :$x^2 \dfrac {\d^2 y} {\d x^2} + x \dfrac {\d y} {\d x} + \paren {x^2 - n^2} y = 0$ These solutions have two main classes: :the Bessel functions of the first kind $J_n$ and: :the Bessel functions of the second kind $Y_n$.


Definition:Monster Group

A group $G$ is a '''Monster group''' and the largest sporadic simple group {{iff}} it has the order:   :$808017424794512875886459904961710757005754368000000000 = 2^{46}.3^{20}.5^9.7^6.11^2.13.17.19.23.29.31.41.47.59.71$


Definition:Dirac Delta Function

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. Consider the real function $F_\epsilon: \R \to \R$ defined as: :$\map {F_\epsilon} x := \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$ The '''Dirac delta function''' is defined as: :$\map \delta x := \displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$


Definition:Symmetric Set/Real Numbers

Let $\R$ be the set of real numbers. Let $S \subseteq \R$ such that: :$\forall x \in S: -x \in S$ That is, for every element in $S$, its negative is also in $S$. Then $S$ is a '''symmetric subset of $\R$''', or (if $\R$ is implicit) $S$ is a '''symmetric set'''.


Definition:Additive Inverse/Ring

Let $\struct {R, +, \circ}$ be a ring whose ring addition operation is $+$. Let $a \in R$ be any arbitrary element of $R$. The '''additive inverse''' of $a$ is its inverse under ring addition, denoted $-a$: :$a + \paren {-a} = 0_R$ where $0_R$ is the zero of $R$.


Definition:Additive Inverse/Number

Let $\Bbb F$ be one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$. Let $a \in \Bbb F$ be any arbitrary number. The '''additive inverse''' of $a$ is its inverse under addition, denoted $-a$: :$a + \paren {-a} = 0$


Definition:Mellin Transform

Let $\map f t: \R_{\ge 0} \to \C$ be a function of a real variable $t$. The '''Mellin Transform''' of $f$, denoted $\map \MM f$ or $\phi$, is defined as: :$\displaystyle \map {\MM \set {\map f t} } s = \map \phi s = \int_0^{\to +\infty} t^{s - 1} \map f t \rd t$ wherever this improper integral exists. Here $\map \MM f$ is a complex function of the variable $s$.


Definition:Quaternion Modulus

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$. Then the '''(quaternion) modulus of''' $\mathbf x$ is written as $\vert \mathbf x \vert$ and is defined as: :$\vert \mathbf x \vert := \sqrt{a^2 + b^2 + c^2 + d^2}$ The '''quaternion modulus''' is a real-valued function, and as when appropriate is referred to as the '''quaternion modulus function'''.


Definition:Quaternion Conjugate/Quaternion Conjugation

The operation of '''quaternion conjugation''' is the mapping: : $\overline \cdot: \mathbb H \to \mathbb H: \mathbf x \mapsto \overline{\mathbf x}$. where $\overline{\mathbf x}$ is the quaternion conjugate of $x$. That is, it maps a quaternion to its quaternion conjugate.


Definition:Natural Numbers/Axiomatization

=== Peano's Axioms === {{:Axiom:Peano's Axioms}} === Naturally Ordered Semigroup === {{:Definition:Naturally Ordered Semigroup}} === 1-Based Natural Numbers === The following axioms are intended to capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$: {{:Axiom:Axiomatization of 1-Based Natural Numbers}}


Definition:Natural Numbers/Construction

=== Von Neumann Construction of Natural Numbers === {{:Definition:Natural Numbers/Von Neumann Construction}} === Inductive Set Definition for Natural Numbers === {{:Definition:Natural Numbers/Inductive Set Definition}} === Inductive Set Definition for Natural Numbers in Real Numbers === {{:Definition:Natural Numbers/Inductive Sets in Real Numbers}}


Definition:Ordering on Natural Numbers

=== Ordering on Peano Structure === {{:Definition:Ordering on Natural Numbers/Peano Structure}} === Ordering on Naturally Ordered Semigroup === {{:Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup}} === Ordering on $1$-Based Natural Numbers === {{:Definition:Ordering on Natural Numbers/1-Based}} === Ordering on Minimal Infinite Successor Set === {{:Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set}} === Ordering on Von Neumann Construction of Natural Numbers === {{:Definition:Ordering on Natural Numbers/Von Neumann Construction}} === Ordering on Natural Numbers in Real Numbers === {{:Definition:Ordering on Natural Numbers/Restriction of Real Numbers}}


Definition:Ordering on Natural Numbers/1-Based

Let $\N_{>0}$ be the axiomatised $1$-based natural numbers. The '''strict ordering of $\N_{>0}$''', denoted $<$, is defined as follows: :$\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$ The '''(weak) ordering of $\N_{>0}$''', denoted $\le$, is defined as: :$\forall a, b \in \N_{>0}: a \le b \iff a = b \lor a < b$


Definition:Ordering on Natural Numbers/Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. The relation $\preceq$ in $\struct {S, \circ, \preceq}$ is called the '''ordering'''.


Definition:Ordering on Natural Numbers/Peano Structure

Let $\struct {P, 0, s}$ be a Peano structure. The '''ordering''' of $P$ is the relation $\le$ defined by: :$\forall m, n \in P: m \le n \iff \exists p \in P: m + p = n$ where $+$ denotes addition in $\struct {P, 0, s}$.


Definition:Ordering on Natural Numbers/Minimal Infinite Successor Set

Let $\omega$ be the minimal infinite successor set. The '''strict ordering''' of $\omega$ is the relation $<$ defined by: :$\forall m, n \in \omega: m < n \iff m \in n$ The '''(weak) ordering''' of $\omega$ is the relation $\le$ defined by: :$\forall m, n \in \omega: m \le n \iff m < n \lor m = n$


Definition:Ordering on Natural Numbers/Restriction of Real Numbers

Let $\struct {\R, +, \times, \le}$ be the field of real numbers. Let $\N$ be the natural numbers in $\R$. Then the '''ordering''' of $\N$ is the restriction of $\le$ to $\N$.


Definition:Addition on 1-Based Natural Numbers

Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $+$ in this axiomatizaton is called '''addition'''.


Definition:Octonion/Addition

The '''sum''' of two octonions $x=\left({a, b}\right)$ and $y=\left({c, d}\right)$ where $a, b, c, d \in \mathbb H$ are quaternions is defined as: :$x + y = \left({a, b}\right) + \left({c, d}\right) = \left({a + c, b + d}\right)$


Definition:Figurate Number

A '''figurate number''' is loosely defined as a (natural) number corresponding to a set of objects which can be arranged in a geometric pattern (in an arbitrary number of dimensions, but usually $2$ or $3$).


Definition:Periodic Continued Fraction/Cycle

The repeating block in a periodic (or purely periodic) continued fraction $F$ is called the '''cycle''' of $F$.


Definition:Theory (Logic)/Complete

$T$ is '''complete''' {{iff}}: : for every $\mathcal L$-sentence $\phi$, either $T \models \phi$ or $T \models \neg \phi$ where $T \models \phi$ denotes semantic entailment.


Definition:Theory (Logic)/Maximal

$T$ is '''maximal''' {{iff}}: :for every $\mathcal L$-sentence $\phi$, either $\phi \in T$ or $\neg \phi \in T$.


Definition:Theory (Logic)/Structure

Let $\mathcal M$ be an $\mathcal L$-structure. The '''$\mathcal L$-theory of $\mathcal M$''' is the $\mathcal L$-theory consisting of those $\mathcal L$-sentences $\phi$ such that: :$\mathcal M \models \phi$ where $\models$ denotes that $\mathcal M$ is a model for $\phi$. This theory can be denoted $\operatorname{Th} \left({\mathcal M}\right)$ when the language $\mathcal L$ is understood.


Definition:Stability (Model Theory)/Kappa-Stable Theory

$T$ is $\kappa$-'''stable''' {{iff}}: : for all models $\mathcal M$ of $T$ : for all subsets $A \subseteq \mathcal M$ of cardinality $\kappa$ and: : for all $n \in \N$ the cardinality $\left\vert{ {S_n}^{\mathcal M} \left({A}\right) }\right\vert$ of the set ${S_n}^{\mathcal M} \left({A}\right)$ of complete $n$-types over $A$ is $\kappa$.


Definition:Stability (Model Theory)/Stable Theory

$T$ is '''stable''' {{iff}} it is $\kappa$-stable for some $\kappa \ge \aleph_0$.


Definition:Stability (Model Theory)/Kappa-Stable Structure

Let $\mathcal M$ be an $\mathcal L$-structure. Let $\operatorname {Th} \left({\mathcal M}\right)$ be the $\mathcal L$-theory of $\mathcal M$. $\mathcal M$ is '''$\kappa$-stable''' if $\operatorname{Th} \left({\mathcal M}\right)$ is $\kappa$-stable.


Definition:Affine Monoid

An '''affine monoid''' is a monoid that is: : finitely generated and: : isomorphic to a submonoid of a free abelian group $\Z^d$, for some $d \in \Z_{\ge 0}$.


Definition:Bounded Linear Transformation/Bounded Linear Operator

Let $H$ be a Hilbert space. Let $A: H \to H$ be a linear operator. $A$ is a '''bounded linear operator''' {{iff}} :$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_H \le c \left\Vert{h}\right\Vert_H$ That is, a '''bounded linear operator''' is a bounded linear transformation from a Hilbert space to itself.


Definition:Arborescence/Root

The vertex $r$ of $G$ is known as the '''root of (the arborescence) $G$'''.


Definition:Arborescence/Also defined as

Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, define an arborescence of root $r$ so as to reverse the orientation of $G$, so that the arcs are all directed toward the root rather than away from it.


Definition:Arborescence/Also known as

An arborescence of root $r$ can be referred to as an an '''$r$-arborescence''', or just an '''arborescence'''. Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, call an arborescence an '''oriented tree'''.


Definition:Interval/Ordered Set/Half-Open

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a, b \in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}}


Definition:Ray (Order Theory)/Open

The following sets are called '''open rays''' or '''open half-lines''': : $\left\{{x \in S: a \prec x}\right\}$ (the strict upper closure of $a$), denoted $a^\succ$ : $\left\{{x \in S: x \prec a}\right\}$ (the strict lower closure of $a$), denoted $a^\prec$.


Definition:Ray (Order Theory)/Closed

The following sets are called '''closed rays''' or '''closed half-lines''': : $\left\{{x \in S: a \preccurlyeq x}\right\}$ (the upper closure of $a$), denoted $a^\succcurlyeq$ : $\left\{{x \in S: x \preccurlyeq a}\right\}$ (the lower closure of $a$), denoted $a^\preccurlyeq$.


Definition:Ray (Order Theory)/Upward-Pointing

An '''upward-pointing ray''' is a ray which is bounded below: : an open ray $a^\succ:= \left\{{x \in S: a \prec x}\right\}$ : a closed ray $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$


Definition:Ray (Order Theory)/Downward-Pointing

A '''downward-pointing ray''' is a ray which is bounded above: : an open ray $a^\prec := \left\{{x \in S: x \prec a}\right\}$ : a closed ray $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$


Definition:Hereditarily Compact Space/Definition 1

$T$ is ''' hereditarily compact''' {{iff}} every subspace of $T$ is compact.


Definition:Hereditarily Compact Space/Definition 2

$T$ is ''' hereditarily compact''' {{iff}}: : for each family $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of open sets of $T$, there exists a finite subset $J \subset I$ such that: :::$\displaystyle \bigcup_{j \mathop \in J} U_j = \bigcup_{i \mathop \in I} U_i$


Definition:Order of Structure/Infinite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be infinite. Then $\struct {S, \circ}$ is an '''infinite structure'''.


Definition:Order of Structure/Finite Structure

Let the underlying set $S$ of $\struct {S, \circ}$ be finite. Then $\struct {S, \circ}$ a '''finite structure'''.


Definition:Index of Subgroup/Finite

If $G / H$ is a finite set, then $\index G H$ is '''finite''', and $H$ is '''of finite index''' in $G$.


Definition:Index of Subgroup/Infinite

If $G / H$ is an infinite set, then $\index G H$ is '''infinite''', and $H$ is '''of infinite index''' in $G$.


Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Infinite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Infinite/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Infinite/Definition 3}}


Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Finite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Finite/Definition 2}}


Definition:Multiplication on 1-Based Natural Numbers

Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $\times$ in this axiomatization is called '''multiplication'''.


Definition:Order of Group Element/Definition 1

The '''order of $x$ (in $G$)''', denoted $\order x$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.


Definition:Order of Group Element/Definition 2

The '''order of $x$ (in $G$)''', denoted $\order x$, is the order of the group generated by $x$: :$\order x := \order {\gen x}$


Definition:Directed Graph/Arc/Endvertex/Final Vertex

The '''final vertex''' of $e$ is the endvertex $v$ which $e$ is incident to.


Definition:Directed Graph/Arc/Endvertex/Initial Vertex

The '''initial vertex''' of $e$ is the endvertex $u$ which $e$ is incident from.


Definition:Integer/Formal Definition/Notation

We have that $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$. As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\eqclass {a, b} \boxminus$, or $\eqclass {a, b} {}$. This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\eqclass {a, b} {}$ at the start of any exposition.


Definition:Injection/Definition 1 a

:$\forall x_1, x_2 \in \Dom f: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$


Definition:Injection/Definition 1

A mapping $f$ is '''an injection''', or '''injective''' {{iff}}: :$\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$ That is, an '''injection''' is a mapping such that the output uniquely determines its input.


Definition:Injection/Definition 2

An '''injection''' is a relation which is both one-to-one and left-total.


Definition:Injection/Definition 3

Let $f$ be a mapping. Then $f$ is an injection {{iff}}: :$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.


Definition:Injection/Definition 4

Let $f$ be a mapping. $f$ is an '''injection''' {{iff}}: :$\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$ where: :$\Img f$ denotes the image set of $f$ :$\card {\, \cdot \,}$ denotes the cardinality of a set :$\map {f^{-1} } y$ is the preimage of $y$ :$f^{-1} \sqbrk {\set y}$ is the preimage of the subset $\set y \subseteq \Img f$.


Definition:Injection/Definition 5

Let $f: S \to T$ be a mapping where $S \ne \O$. Then $f$ is an injection {{iff}}: :$\exists g: T \to S: g \circ f = I_S$ where $g$ is a mapping. That is, {{iff}} $f$ has a left inverse.


Definition:Injection/Definition 6

Let $f: S \to T$ be a mapping where $S \ne \O$. Then $f$ is an '''injection''' {{iff}} $f$ is left cancellable: :$\forall X: \forall g_1, g_2: X \to S: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ where $g_1$ and $g_2$ are arbitrary mappings from an arbitrary set $X$ to the domain $S$ of $f$.


Definition:Associative Operation

Let $S$ be a set. Let $\circ : S \times S \to S$ be a binary operation. Then $\circ$ is '''associative''' {{iff}}: :$\forall x, y, z \in S: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$


Definition:Set/Uniqueness of Elements

A set is '''uniquely determined''' by its elements. This means that the only thing that defines '''what a set is''' is '''what it contains'''. So, how you choose to '''list''' or '''define''' the contents makes '''no difference''' to what the contents actually '''are'''.


Definition:Tally Marks

'''Tally marks''' are a system of recording (usually) small numbers. For each unit in a given natural number, a mark is made. Thus, for example, $13$ could be recorded $|||||||||||||$. It is usual for '''tally marks''' to be arranged in groups of $5$, by drawing every fifth line diagonally across the group of $4$ before it. Thus $13$ would then be recorded: :File:Tally5.pngFile:Tally5.pngFile:Tally3.png


Definition:Lebombo Bone

The '''Lebombo bone''' is a baboon's leg bone bearing $29$ notches which were carved into it between $33 \ 000$ B.C.E. and $42 \ 000$ B.C.E. It is the earliest known instance of a system of tally marks. It was found in {{WP|Border_Cave|Border Cave}} in {{WP|South_Africa|South Africa}} in the {{WP|Lebombo_Mountains|Lebombo mountains}}, near the border with {{WP|Swaziland|Swaziland}}. Various untestable hypotheses have been raised concerning this artefact, notably that it was created by an early African woman to count the days between her menses. However, as the bone appears to be broken at one end, it is possible that the count of $29$ is not the complete set of notches that were originally carved.


Definition:Wolf Bone

The '''(Czechoslovakian) wolf bone''' is a wolf bone dating from approximately $30 \ 000$ years ago, bearing $57$ marks which appear to have been made deliberately. It was discovered in $1937$ in {{WP|Czechoslovakia|Czechoslovakia}} at {{WP|Doln%C3%AD_V%C4%9Bstonice_(archaeology)|Dolní Věstonice}} in {{WP|Moravia|Moravia}}, by a team led by {{WP|Karel_Absolon|Karl Absolon}}. The markings are believed to be tally marks, and are arranged in $11$ groups of $5$, with $2$ left over. The untestable hypothesis has been made that the number corresponds to a double lunar period.


Definition:Ishango Bone

The '''Ishango bone''' is a mathematical tool made from a leg bone of a baboon, dating from approximately $25 \ 000$ years ago. It bears a system of markings which may be tally marks, but may in fact be for a more complex purpose. It was found in $1960$ by {{WP|Jean_de_Heinzelin_de_Braucourt|Jean de Heinzelin de Braucourt}} in the {{WP|Belgian_Congo|Belgian Congo}} (now the {{WP|Democratic_Republic_of_the_Congo|Democratic Republic of the Congo}}). It was discovered in the area of {{WP|Ishango|Ishango}} near the {{WP|Semliki_River|Semliki River}}. One row of marks contains the prime numbers between $10$ and $20$, that is, $11$, $13$, $17$ and $19$, which total $60$. Another row contains $9$, $11$, $19$ and $21$, which also total $60$. The third row contains marks which appear to represent a method of multiplication by means of a technique involving doubling and halving. While it is suggested that the markings form a calendar, it is also possible that the markings are coincidental, and perhaps even just put there to make the bone easier to hold.


Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by $I$. Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $S$ be a set such that: :$\forall i \in I: S_i = S$ === Definition 1 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2}}


Definition:Ordered Tuple/Definition 1

An '''ordered tuple (of length $n$)''' is a finite sequence whose domain is $\N^*_n$.


Definition:Ordered Tuple/Also known as

Some sources refer to an ordered tuple as a '''tuple'''. The term '''ordered $n$-tuple''' can sometimes be seen, particularly for specific instances of $n$.  Instead of writing '''2-tuple''', '''3-tuple''' and '''4-tuple''', the terms '''couple''', '''triple''' and '''quadruple''' are usually used. In the context of abstract algebra, the concept is encountered as '''(associative) word'''.


Definition:Ordered Tuple/Notation

Notation for an ordered tuple varies throughout the literature. There are also specialised instances of an ordered tuple where the convention is to use angle brackets. However, it is common for an ordered tuple to be denoted: :$\tuple {a_1, a_2, \ldots, a_n}$ extending the notation for an ordered pair. For example: $\tuple {6, 3, 3}$ is the ordered triple $f$ defined as: :$\map f 1 = 6, \map f 2 = 3, \map f 3 = 3$ The notation: :$\sequence {a_1, a_2, \ldots, a_n}$ is recommended when use of round brackets would be ambiguous. Other notations which may be encountered are: :$\sqbrk {a_1, a_2, \ldots, a_n}$ :$\set {a_1, a_2, \ldots, a_n}$ but both of these are strongly discouraged: the square bracket format because there are rendering problems on this site, the latter because it is too easily confused with set notation. In order to further streamline notation, it is common to use the more compact $\sequence {a_n}$ for $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$. Some sources, particularly in such fields as communication theory, where the elements of the domain of the ordered tuple is a specific set of symbols, use the notation $x_1 x_2 \cdots x_n$ for $\tuple {x_1, x_2, \dotsc, x_n}$.


Definition:Ordered Tuple/Definition 2

Let $\family {S_i}_{i \mathop \in \N_n}$ be a family of sets indexed by $\N_n$. Let $\displaystyle \prod_{i \mathop \in \N_n} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in \N_n}$. An '''ordered tuple of length $n$''' of $\family {S_i}$ is an element of $\displaystyle \prod_{i \mathop \in \N_n} S_i$.


Definition:Direct Image Mapping/Mapping

Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. The '''direct image mapping''' of $f$ is the mapping $f^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq S$ to its image under $f$: :$\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$


Definition:Direct Image Mapping/Relation

Let $\RR \subseteq S \times T$ be a relation on $S \times T$. The '''direct image mapping''' of $\RR$ is the mapping $\RR^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq T$ to its image under $\RR$: :$\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} & : X \ne \O \\ \O & : X = \O \end {cases}$


Definition:Inverse Image Mapping

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. === Relation === Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. {{Definition:Inverse Image Mapping/Relation/Definition 1}} === Mapping === Let $f: S \to T$ be a mapping. {{Definition:Inverse Image Mapping/Mapping/Definition 1}}


Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. === Definition 1 === {{:Definition:Inverse Image Mapping/Relation/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Relation/Definition 2}}


Let $f: S \to T$ be a mapping. === Definition 1 === {{:Definition:Inverse Image Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Mapping/Definition 2}}


Definition:Equivalence Relation/Also denoted as

When discussing equivalence relations, various notations are used for $\tuple {x, y} \in \RR$. Examples are: :$x \mathrel \RR y$ :$x \equiv \map y \RR$ :$x \equiv y \pmod \RR$ and so on. Specialised equivalence relations generally have their own symbols, which can be defined as they are needed. Such symbols include: :$\cong$, $\equiv$, $\sim$, $\simeq$, $\approx$


Definition:Equivalence Relation/Definition 2

$\RR$ is an equivalence relation {{iff}}: :$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$ where: :$\Delta_S$ denotes the diagonal relation on $S$ :$\RR^{-1}$ denotes the inverse relation :$\circ$ denotes composition of relations


Definition:Equivalence Relation/Definition 1

Let $\RR$ be: :$(1): \quad$ reflexive :$(2): \quad$ symmetric :$(3): \quad$ transitive Then $\RR$ is an '''equivalence relation''' on $S$.


Definition:Minimal/Ordered Set/Definition 1

An element $x \in T$ is a '''minimal element of $T$''' {{iff}}: :$\forall y \in T: y \preceq x \implies x = y$ That is, the only element of $T$ that $x$ '''succeeds or is equal to''' is itself.


Definition:Minimal/Ordered Set/Definition 2

An element $x \in T$ is a '''minimal element of $T$''' {{iff}}: :$\neg \exists y \in T: y \prec x$ where $y \prec x$ denotes that $y \preceq x \land y \ne x$. That is, {{iff}} $x$ has '''no strict predecessor'''.


Definition:Minimal/Ordered Set/Comparison with Smallest Element

Compare the definition of minimal element with that of a smallest element. An element $x \in T$ is '''the''' smallest element of $T$ {{iff}}: :$\forall y \in T: x \preceq y$ That is, $x$ is comparable to, and precedes, or is equal to, every $y \in T$. An element $x \in T$ is '''a''' minimal element of $T$ {{iff}}: :$y \preceq x \implies x = y$ That is, $x$ precedes, or is equal to, every $y \in T$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a minimal element is indeed '''the smallest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''minimal element''' and '''smallest element''' are equivalent.


Definition:Maximal/Ordered Set/Definition 1

An element $x \in T$ is a '''maximal element of $T$''' {{iff}}: :$x \preceq y \implies x = y$ That is, the only element of $S$ that $x$ '''precedes or is equal to''' is itself.


Definition:Maximal/Ordered Set/Definition 2

An element $x \in T$ is a '''maximal element of $T$''' {{iff}}: :$\neg \exists y \in T: x \prec y$ where $x \prec y$ denotes that $x \preceq y \land x \ne y$. That is, {{iff}} $x$ has '''no strict successor'''.


Definition:Maximal/Ordered Set/Comparison with Greatest Element

Compare the definition of maximal element with that of a greatest element. An element $x \in T$ is '''the''' greatest element of $T$ {{iff}}: :$\forall y \in T: y \preceq x$ That is, $x$ is comparable to, and succeeds, or is equal to, every $y \in S$. An element $x \in S$ is '''a''' maximal element of $T$ {{iff}}: :$x \preceq y \implies x = y$ That is, $x$ succeeds, or is equal to, every $y \in S$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a maximal element is indeed '''the greatest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''maximal element''' and '''greatest element''' are equivalent.


Definition:Choice Function/Use of Axiom of Choice

The Axiom of Choice (abbreviated '''AoC''' or '''AC''') is the following statement: :''All $\mathbb S$ as above have a choice function.'' It can be shown that the AoC it does not follow from the other usual axioms of set theory, and that it is relative consistent to these axioms (i.e., that AoC does not make the axiom system inconsistent, provided it was consistent without AoC). Note that for any given set $S \in \mathbb S$, one can select an element from it (without using AoC). AoC guarantees that there is a choice function, i.e., a function that "simultaneously" picks elements of all $S \in \mathbb S$.   AoC is needed to prove statements such as "all countable unions of finite sets are countable" (for many specific such unions this  can be shown without AoC), and AoC is equivalent to many other mathematical statements such as "every vector space has a basis".


Definition:Bessel Function/First Kind

A '''Bessel function of the first kind of order $n$''' is a Bessel function which is non-singular at the origin It is usually denoted $\map {J_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\map {J_n} x$ forms a solution.


Definition:Bessel Function/Second Kind

A '''Bessel function of the second kind of order $n$''' is a Bessel function which is singular at the origin. It is usually denoted $\map {Y_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\map {Y_n} x$ forms a solution.


Definition:Pythagoreans

The '''Pythagoreans''' were a semi-mystical cult which dated from around $550$ B.C.E., founded by {{AuthorRef|Pythagoras of Samos}}. Can claim to be the world's first university. It is feasible to suggest that their initial work in the field of geometry may have formed the basis of at least the first two books of {{ElementsLink}}. Attendees of the school were divided into two classes: :the '''Probationers''' (or '''listeners''') :the '''Pythagoreans'''. A student was a '''listener''' for $3$ years, after which he was allowed to be initiated into the class of '''Pythagoreans''', who were allowed to learn what was considered to be the deeper secrets. '''Pythagoreans''' were a closely-knit brotherhood, who held all their worldly goods in common, and were bound by oath never to reveal the secrets of the Founder. There exists a legend that one of the '''Pythagoreans''' was thrown overboard to drown after having revealed the existence of the regular dodecahedron. For some considerable time they dominated the political life of {{WP|Crotone|Croton}}, where they were based, but in $501$ B.C.E. there was a popular revolt in which a number of the leaders of the school were murdered. {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself was murdered soon after. Some sources state that the reasons for this were based on the fact that their puritanical philosophy was at odds with the contemporary way of thinking. Others suggest that there was a reaction against their autocratic rule. Whatever the truth of the matter, between $501$ and about $460$ B.C.E. the political influence of the cult was destroyed. Its survivors scattered, many of them fleeing to {{WP|Thebes,_Egypt|Thebes}} in {{WP|Upper_Egypt|Upper Egypt}}, where they continued to exist more as a philosophical and mathematical society for another couple of centuries, secretive and ascetic to the end, publishing nothing, ascribing all their discoveries to the Master, {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself. === Quadrivium === {{:Definition:Pythagoreans/Quadrivium}} === Trivium === {{:Definition:Pythagoreans/Trivium}}


Definition:Method of Exhaustion

The '''method of exhaustion''' is a technique for calculating an approximation to the physical extent of a geometric figure whose extremities are curved lines. To measure the area of a circle, for example, this is done by: : Circumscribing a polygon around the circle : Inscribing a polygon inside the circle : Measuring the areas of those polygons : Noting that the area of the circle is between those two : Increasing the number of sides on those polygons to make them closer and closer to the circle and each other. As the number of sides of the polygons increases, their areas become ever closer to each other.


Definition:Markov Chain

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a sequence of random variables in a countable set $S$. Let $\map \Pr X$ denote the probability of the random variable $X$. Let $\sequence {X_n}_{n \mathop \ge 0}$ satisfy the Markov property: :$\map \Pr {X_{n + 1} = i_{n + 1} \mid X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n} = \map \Pr {X_{n + 1} = i_{n + 1} \mid X_n = i_n}$ for all $n \ge 0$ and all $i_0, i_1, \ldots, i_{n + 1} \in S$. Then $\sequence {X_n}_{n \mathop \ge 0}$ is a '''Markov chain'''.


Definition:Markov Chain/State Space

The set $S$ is called the '''state space''' of the Markov chain.


Definition:Tool

A '''tool''' is a physical device used as an aid to a process.


Definition:Straightedge

A '''straightedge''' is an ideal tool for constructing straight lines. A '''straightedge''' is of unlimited length, but has no markings on it, so it cannot be used for measurement.


Definition:Compass

A '''compass''' is an ideal tool for drawing circles. Hence it can be used according to Euclid's third postulate to construct a circle using any two given points: :$(1): \quad$ the center and: :$(2): \quad$ an arbitrary point on the circumference.


Definition:Compass and Straightedge Construction

A '''compass and straightedge construction''' is a technique of drawing geometric figures using only a straightedge and a compass. The operations available are: :using the straightedge to draw a straight line determined by two given points :using the compass to draw a circle whose center is at a given point and whose radius is the distance between two given points :finding the points of intersection between straight lines and circles.


Definition:Pólya's Urn

'''Pólya's urn''' is an implementation of an urn such that, when a ball of a particular colour is taken out and examined, that ball is replaced and another of the same colour is added. {{NamedforDef|George Pólya|cat = Pólya}} Category:Definitions/Probability Theory ncbllm09u1lm8ewv2p4t2zx98p91h4n


Definition:Trisection

To '''trisect''' a finite geometrical object is to cut it into $3$ equal parts.


Definition:Neusis Ruler

A '''neusis ruler''' is a straightedge on which two points have been marked with a specific fixed length between them.


Definition:Neusis Construction

A '''neusis construction''' is a technique of drawing geometric figures using only a neusis ruler and a compass.


Definition:Neusis Ruler/Diastema

On a neusis ruler, the '''diastema''' is the specific fixed length between the two marked points.


Definition:Line/Curve/Plane

A '''plane curve''' is a curve which can be embedded in the plane.


Definition:Conic Section/Focus-Directrix Property

A '''conic section''' is a plane curve which can be specified in terms of: :a given straight line $D$ known as the directrix :a given point $F$ known as a focus :a given constant $\epsilon$ known as the eccentricity. Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition: :$(1): \quad q = \epsilon \, p$ Then $K$ is a '''conic section'''. Equation $(1)$ is known as the '''focus-directrix property''' of $K$.


Definition:Conic Section/Intersection with Cone

:600px Let $C$ be a double napped right circular cone whose base is $B$. Let $\theta$ be half the opening angle of $C$. That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$. Let a plane $D$ intersect $C$. Let $\phi$ be the inclination of $D$ to the axis of $C$. Let $K$ be the set of points which forms the intersection of $C$ with $D$. Then $K$ is a '''conic section''', whose nature depends on $\phi$.


Definition:Parabola/Focus-Directrix

:300px Let $D$ be a straight line. Let $F$ be a point. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$: :$p = q$ Then $K$ is a '''parabola'''.


Definition:Hyperbola/Focus-Directrix

:300px Let $D$ be a straight line. Let $F_1$ be a point. Let $\epsilon \in \R: \epsilon > 1$. Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F_1$ are related by the condition: :$\epsilon \, p = q$ Then $K$ is a '''hyperbola'''.


Definition:Conic Section/Directrix

The line $D$ is known as the '''directrix''' of the conic section.


Definition:Conic Section/Focus

The point $F$ is known as the '''focus''' of the conic section.


Definition:Conic Section/Eccentricity

The constant $\epsilon$ is known as the '''eccentricity''' of the conic section.


Definition:Ellipse/Focus

The point $F$ is known as the '''focus''' of the ellipse.


Definition:Ellipse/Directrix

The line $D$ is known as the '''directrix''' of the ellipse.


Definition:Ellipse/Eccentricity

The constant $\epsilon$ is known as the '''eccentricity''' of the ellipse.


Definition:Hyperbola/Focus

The point $F_1$ is known as a '''focus''' of the hyperbola. The symmetrically-positioned point $F_2$ is also a '''focus''' of the hyperbola.


Definition:Hyperbola/Directrix

The line $D$ is known as the '''directrix''' of the hyperbola.


Definition:Hyperbola/Eccentricity

The constant $\epsilon$ is known as the '''eccentricity''' of the hyperbola.


Definition:Parabola/Focus

The point $F$ is known as the '''focus''' of the parabola.


Definition:Parabola/Directrix

The line $D$ is known as the '''directrix''' of the parabola.


Definition:Greek Numerals

=== Attic System === {{:Definition:Greek Numerals/Attic System}} === Classical Period === {{:Definition:Greek Numerals/Classical Period}}


Definition:Greek Numerals/Attic System

The '''Greek numerals''' from the Attic period (c. 500 BCE to c. 300 BCE) are similar to the Roman system: {{begin-eqn}} {{eqn | l = \textsf{I}:       | o =        | r = \)one\( }} {{eqn | l = \Pi:       | o = \)'''penta'''\(:       | r = \)five\( }} {{eqn | l = \Delta:       | o = \)'''deka'''\(:       | r = \)ten\( }} {{eqn | l = \Eta:       | o = \)'''hekaton'''\(:       | r = \)one hundred\( }} {{eqn | l = \Xi:       | o = \)'''chilioi'''\(:       | r = \)one thousand\( }} {{eqn | l = \Mu:       | o = \)'''myriad'''\(:       | r = \)ten thousand\( }} {{end-eqn}} $\Pi$ was later written in a different form, with a shorter right leg, and from there it appears to have evolved into $\Gamma$. Numbers were combined in a similar way to Roman numbers, but with additive forms only. For example, $32718$ would have been written $\Mu \Mu \Mu \Xi \Xi \Eta \Eta \Eta \Eta \Eta \Eta \Eta \Delta \Pi \textsf {III}$. Later evolutions introduced symbols for $50$, $500$ and $5000$, consisting of a tiny version of the appropriate power of $10$ nestled under the right hand (short) branch of the short-leg version of $\Pi$. Such characters are difficult to render neatly, and as they are of limited importance, this has not been attempted on {{ProofWiki}}.


Definition:Greek Numerals/Classical Period

The '''Greek numerals''' from the Classical period (c. $600$ BCE to c. $300$ BCE) are as follows: :$\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \alpha & \beta & \gamma & \delta & \epsilon & stigma & \zeta & \eta & \theta \\ \hline \\ 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\ \iota & \kappa & \lambda & \mu & \nu & \xi & \omicron & \pi & koppa \\ \hline \\ 100 & 200 & 300 & 400 & 500 & 600 & 700 & 800 & 900 \\ \rho & \sigma & \tau & \upsilon & \phi & \chi & \psi & \omega & sampi \\ \end{array}$ where $stigma$, $koppa$ and $sampi$ are characters that are not supported by MathJax. These were the letters of the everyday Greek alphabet, with extra ones added from the Phoenician alphabet. As it could be easy to confuse numbers with the letters of Greek words, a line was written over the numbers to distinguish them. Numbers bigger than $999$ could be written by putting a stroke in front of the symbols.


Definition:Rational Number/Fraction/Denominator

The term $b$ is known as the '''denominator''' of $\dfrac a b$.


Definition:Rational Number/Fraction/Numerator

The term $a$ is known as the '''numerator''' of $\dfrac a b$.


Definition:Trigonometry

'''Trigonometry''' is the branch of mathematics which studies the relationships between the lengths of sides and angles of triangles.


Definition:Spherical Geometry

'''Spherical geometry''' is the branch of mathematics which concerns spatial relationships on the surface of a sphere.


Definition:Spherical Trigonometry

'''Spherical trigonometry''' is the branch of mathematics which concerns the measurement of spherical triangles and related figures on the surface of a sphere.


Definition:Napier's Bones

rightNapier's Rods '''Napier's bones''' is a device for performing the operation of multiplication of numbers. The device consists of: : a set of '''rods''', one for each digit $0$ to $9$, containing the multiples of that number, again from $0$ to $9$, in a column, written as $2$ digits, divided by a diagonal line. : a '''board''' upon which the '''rods''' can be placed, labeled $1$ to $9$ down a column on the left hand side, each digit corresponding to the numbers on the '''rods''' The multiplicand is assembled by placing the rods for each of its digit next to each other on the '''board'''. The multiplier corresponds to the number down the left hand side of the '''board''' For each digit in the multiplicand, the numbers adjacent to each other on the row corresponding to the multiplier in each of the compartments formed by the diagonal lines aligned together are added together. The product is thus formed.


Definition:Prosthaphaeresis

'''Prosthaphaeresis''' is the process whereby multiplication and division are performed by: :$(1): \quad$ converting the numbers into a form where they can be added or subtracted :$(2): \quad$ performing the addition or subtraction :$(3): \quad$ converting the result back again, using the inverse process to step $(1)$.


Definition:Prosthaphaeresis/Trigonometry

The trigonometric technique of prosthaphaeresis is based on the Prosthaphaeresis Formula for Sine plus Sine: :$\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$


Definition:Napierian Logarithm

The '''Napierian logarithms''' are a system of logarithms where: :$\log_{\text {Nap} } 10 \, 000 \, 000 = 0$ :$\log_{\text {Nap} } 9 \, 999 \, 999 = 1$ Thus: :$\log_{\text {Nap} } 10^7 x y = \log_{\text {Nap} } x + \log_{\text{Nap} } y$


Definition:Prosthaphaeresis/Logarithms

The logarithmic technique of prosthaphaeresis is based on the rule of the Sum of Logarithms: :$\log_b x y = \log_b x + \log_b y$ Any base $b$ can be used. The original tecnhique used Napierian logarithms where the base was $0.9999999$ In due course the common (base $10$) logarithm was used, as developed by {{AuthorRef|Henry Briggs}}, called Briggsian logarithms.


Definition:Epicycle

An '''epicycle''' is the path described by an object moving in a uniform circular motion around a point which is itself moving in a uniform circular motion around another point. :400px That point may itself also be moving in a uniform circular motion around yet another point.


Definition:Oblique Axes

'''Oblique axes''' are a pair of coordinate axes which are not at right angles to each other. :400px


Definition:Equation of Geometric Figure

Let there be a coordinate system. Let the variables $\alpha, \beta, \gamma, \ldots$ be used to identify points within that coordinate system. Let the variables be related to each other by means of an equation. Then the equations define the set of points which form a geometric figure. This equation is known as the '''equation of the geometric figure'''.


Definition:Tensor Product of Modules

=== Commutative ring === Let $R$ be a commutative ring with unity. Let $M$ and $N$ be $R$-modules. === Definition 1 === Their '''tensor product''' is a pair $(M \otimes_R N, \theta)$ where: :$M \otimes_R N$ is an $R$-module :$\theta : M \times N \to M \otimes_R N$ is an $R$-bilinear mapping satisfying the following '''universal property''': :For every pair $(P, \omega)$ of an $R$-module and an $R$-bilinear mapping $\omega : M \times N \to P$, there exists a unique $R$-module homomorphism $f : M \otimes_R N \to P$ with $\omega = f \circ \theta$. === Definition 2 === Their '''tensor product''' is the pair $\struct {M \otimes_R N, \theta}$, where: :$M \otimes_R N$ is the quotient of the free $R$-module $R^{\paren {M \times N} }$ on the direct product $M \times N$, by the submodule generated by the set of elements of the form: ::$\tuple {\lambda m_1 + m_2, n} - \lambda \tuple {m_1, n} - \tuple {m_2, n}$ ::$\tuple {m, \lambda n_1 + n_2} - \lambda \tuple {m, n_1} - \tuple {m, n_2}$ ::for $m, m_1, m_2 \in M$, $n, n_1, n_2 \in N$ and $\lambda \in R$, where we denote $tuple {m, n}$ for its image under the canonical mapping $M \times N \to R^{\paren {M \times N} }$. :$\theta : M \times N \to M \otimes_R N$ is the composition of the canonical mapping $M \times N \to R^{\paren {M \times N} }$ with the quotient module homomorphism $R^{\paren {M \times N} } \to M \otimes_R N$. === Noncommutative ring === Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. First construct a left module as a direct sum of all free left modules with a basis that is a single ordered pair in $M \times N$ which is denoted $\map R {m, n}$. :$T = \displaystyle \bigoplus_{s \mathop \in M \mathop \times N} R s$ That this is indeed a module is demonstrated in Tensor Product is Module. Next for all $m, m' \in M$, $n, n' \in N$ and $r \in R$ we construct the following free left modules. :$L_{m, m', n}$ with a basis of $\tuple {m + m', n}$, $\tuple {m, n}$ and $\tuple {m', n}$ :$R_{m, n, n'}$ with a basis of $\tuple {m, n + n'}$, $\tuple {m, n}$ and $\tuple {m, n'}$ :$A_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m r, n}$ :$B_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m, r n}$ Let: :$D = \displaystyle \map {\bigoplus_{r \in R, n, n' \in N, m, m' \in M} } {L_{m, m', n} \oplus R_{m, n, n'} \oplus A_{r, m, n} \oplus B_{r, m, n} }$ The '''tensor product''' $M \otimes_R N$ is then our quotient module $T / D$.


Definition:Folium of Descartes

=== Cartesian Form === {{:Definition:Folium of Descartes/Cartesian Form}} === Parametric Form === {{:Definition:Folium of Descartes/Parametric Form}} :500px


Definition:Archimedean Spiral

The '''Archimedean spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = a \theta$ :600px


Definition:Number Theory

'''Number theory''' is the branch of mathematics which studies the properties of the natural numbers.


Definition:Algebraic Number Theory

'''Algebraic number theory''' is the branch of abstract algebra which studies structures in which the usual number fields are embedded. As such it can also be considered to be a branch of number theory.


Definition:Topology (Mathematical Branch)

'''Topology''' is a geometry of transformations in which the only invariant is continuity. Some sources suggest that it can indeed be described simply as '''the study of continuity'''. As such, it is closely interwoven with the branch of '''analysis'''.


Definition:Plane Geometry

'''Plane geometry''' is the study of geometric figures in two dimensions.


Definition:Computer Science

'''Computer science''' is the branch of mathematics studying questions arising from the operation of [https://en.wikipedia.org/wiki/Computer digital computers]. As such it can be considered as an offshoot of both mathematical logic and discrete mathematics.


Definition:Computability Theory

'''Computability theory''' is a branch of mathematical logic which concerns itself with the algorithmic implementation of mathematical proofs.


Definition:Functional Analysis

'''Functional analysis''' is a branch of analysis, which studies vector spaces endowed a structure such as inner product, norm or topology.


Definition:Analysis

'''Analysis''' is a branch of mathematics that studies continuous change. It subsumes the fields of calculus, differential equations, the calculus of variations, power series, Fourier series, real analysis and complex analysis. It is one of the main branches of mathematics, alongside geometry, number theory and algebra.


Definition:Analysis/Real

'''Real analysis''' is a branch of mathematics that studies real functions.


Definition:Analysis/Complex

'''Complex analysis''' is a branch of mathematics that studies complex functions.


Definition:Ring Theory

'''Ring Theory''' is a branch of abstract algebra which studies rings and other related algebraic structures.


Definition:Analytic Number Theory

'''Analytic number theory''' is a branch of number theory which uses tools from analysis to solve problems about the integers.


Definition:Calculus/Differential

'''Differential calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities change.


Definition:Calculus/Integral

'''Integral calculus''' is a subfield of calculus which is concerned with the study of the rates at which quantities accumulate. Equivalently, given the rate of change of a quantity '''integral calculus''' provides techniques of providing the quantity itself. The equivalence of the two uses are demonstrated in the Fundamental Theorem of Calculus. The technique is also frequently used for the purpose of calculating areas and volumes of curved geometric figures.


Definition:Calculus

'''Calculus''' is the branch of mathematics which studies change.


Definition:Mathematical Physics

'''Mathematical physics''' is the branch of mathematics concerned with the development of mathematical methods for application to problems in physics.


Definition:Physics

'''Physics''' is the branch of science which studies basic concepts including motion, energy, force, and the fundamental nature of spacetime. The study of '''physics''' is generally based in mathematical concepts, especially those of calculus.


Definition:Knot Theory

'''Knot theory''' is the branch of topology which studies the embedding of knots in $3$-dimensional space.


Definition:Calculus of Variations

'''Calculus of variations''' is the subfield of analysis concerned maximizing or minimizing real functionals, which are mappings from a set of functions to the real numbers.


Definition:Matrix Algebra

'''Matrix algebra''' is the subfield of algebra whose subjects of manipulation are matrices.


Definition:Graph Theory

'''Graph theory''' is the branch of mathematics concerned with the structure and properties of graphs. As a (graph-theoretical) graph has the same conceptual definition as a relation, it follows that there is considerable overlap between the fields of graph theory and relation theory.


Definition:Relation Theory

'''Relation theory''' is the subfield of set theory concerned with the properties of relations and relational structures. As a relation has the same conceptual definition as a (graph-theoretical) graph, it follows that there is considerable overlap between the fields of relation theory and graph theory.


Definition:Mapping Theory

'''Mapping theory''' is the subfield of set theory concerned with the properties of mappings.


Definition:Measure Theory

'''Measure theory''' is the subfield of analysis concerned with the properties of measures, particularly the Lebesgue measure.


Definition:Linear Algebra

'''Linear algebra''' is the branch of algebra which studies vector spaces and linear transformations between them.


Definition:Commutative Algebra (Mathematical Branch)

'''Commutative algebra''' is the branch of abstract algebra concerned with commutative and unitary rings.


Definition:Hamiltonian Mechanics

'''Hamiltonian mechanics''' was a reformulation of classical mechanics using a more abstract mathematical framework. It laid the groundwork for quantum mechanics.


Definition:Classical Mechanics

'''Classical mechanics''' is the study of the physical laws describing the motion of and forces on bodies under the action of a system of forces.


Definition:Galois Theory

'''Galois theory''' is a subfield of abstract algebra which provides a connection between field theory and group theory.


Definition:Field Theory

'''Field Theory''' is a branch of abstract algebra which studies fields and other related algebraic structures.


Definition:Lattice Theory

'''Lattice theory''' is the branch of abstract algebra concerned with the structure and properties of lattices.


Definition:Geometrical Mechanics

'''Geometrical mechanics''' is the branch of mathematics which uses techniques from geometry to solve problems in mechanics.


Definition:Mechanics

'''Mechanics''' is the branch of applied mathematics concerned with the motion of and forces on bodies.


Definition:Dimensional Analysis

'''Dimensional analysis''' is the branch of applied mathematics which studies the relationships between physical quantities by identifying their fundamental dimensions and units of measurement. As such it can also be considered to be a branch of number theory.


Definition:Applied Mathematics

'''Applied mathematics''' is the branch of mathematics which concerns the solution of real world applications by the construction of mathematical models.


Definition:Astronomy

'''Astronomy''' is the branch of physics which studies the features which can be seen in the sky that are outside the atmosphere of Earth. Certain of its mathematical aspects can be considered a subfield of solid geometry.


Definition:Modulo Arithmetic

'''Modulo arithmetic''' is the branch of abstract algebra which studies the residue class of integers under a modulus. As such it can also be considered to be a branch of number theory.


Definition:Homological Algebra

'''Homological algebra''' is the branch of algebra and topology which studies homology.


Definition:Umbral Calculus

'''Umbral calculus''' is a branch of analysis which studies polynomial spaces by means of linear functionals. The modern approach concerns the study of Sheffer sequences


Definition:Order Theory

'''Order theory''' is the branch of relation theory which studies orderings.


Definition:Category Theory

'''Category theory''' is the branch of abstract algebra which studies categories. It can be described as the '''theory of functors'''.