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| In [[mathematics]], the composition of [[binary relation]]s is a concept of forming a new relation {{nowrap|''S'' ∘ ''R''}} from two given relations ''R'' and ''S'', having as its most well-known special case the [[composition of functions]].
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| == Definition ==
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| If <math>R\subseteq X\times Y</math> and <math>S\subseteq Y\times Z</math> are two binary relations, then
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| their composition <math>S\circ R</math> is the relation
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| :<math>S\circ R = \{ (x,z)\in X\times Z\mid \exists y\in Y: (x,y)\in R\land (y,z)\in S \}.</math>
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| In other words, <math>S\circ R\subseteq X\times Z</math> is defined by the rule that says <math>(x,z)\in S\circ R</math> if and only if there is an element <math>y\in Y</math> such that <math>x\,R\,y\,S\,z</math> (i.e. <math>(x,y)\in R</math> and <math>(y,z)\in S</math>).
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| In particular fields, authors might denote by {{nowrap|''R'' ∘ ''S''}} what is defined here to be {{nowrap|''S'' ∘ ''R''}}.
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| The convention chosen here is such that [[function composition]] (with the usual notation) is obtained as a special case, when ''R'' and ''S'' are [[functional relation]]s. Some authors<ref>Kilp, Knauer & Mikhalev, p. 7</ref> prefer to write <math>\circ_l</math> and <math>\circ_r</math> explicitly when necessary, depending whether the left or the right relation is the first one applied.
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| A further variation encountered in computer science is the [[Z notation]]: <math>\circ</math> is used to denote the traditional (right) composition, but ⨾ <span style="font-size:180%; font-weight: 600;">;</span> (a fat open semicolon with Unicode code point U+2A3E<ref>http://www.fileformat.info/info/unicode/char/2a3e/index.htm</ref>) denotes left composition. This use of semicolon [[Function_composition#Alternative_notations|coincides]] with the notation for function composition used (mostly by computer scientists) in [[Category theory]], as well as the notation for dynamic conjunction within linguistic dynamic semantics.<ref>http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog</ref>
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| The binary relations <math>R\subseteq X\times Y</math> are sometimes regarded as the morphisms <math>R\colon X\to Y</math> in a [[category (mathematics)|category]] '''[[Category of relations|Rel]]''' which has the sets as objects. In '''Rel''', composition of morphisms is exactly composition of relations as defined above. The category '''[[category of sets|Set]]''' of sets is a subcategory of '''Rel''' that has the same objects but fewer morphisms. A generalization of this is found in the theory of [[allegory (category theory)|allegories]].
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| ==Properties==
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| Composition of relations is [[associative]].
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| The [[inverse relation]] of {{nowrap| ''S'' ∘ ''R''}} is
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| {{nowrap|1= (''S'' ∘ ''R'')<sup>-1</sup> = ''R''<sup>−1</sup> ∘ ''S''<sup>−1</sup>}}. This property makes the set of all binary relations on a set a [[semigroup with involution]].
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| The compose of [[partial function|(partial) function]]s (i.e. functional relations) is again a (partial) function.
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| If ''R'' and ''S'' are [[injective]], then {{nowrap| ''S'' ∘ ''R''}} is injective, which conversely implies only the injectivity of ''R''.
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| If ''R'' and ''S'' are [[surjective]], then {{nowrap| ''S'' ∘ ''R''}} is surjective, which conversely implies only the surjectivity of ''S''.
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| The set of binary relations on a set ''X'' (i.e. relations from ''X'' to ''X'') together with (left or right) relation composition forms a [[monoid]] with zero, where the identity map on ''X'' is the [[neutral element]], and the empty set is the [[Absorbing element|zero element]].
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| == Join: another form of composition ==
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| {{main|Join (relational algebra)}}
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| Other forms of composition of relations, which apply to general ''n''-place relations instead of binary relations, are found in the ''[[Join (relational algebra)|join]]'' operation of [[relational algebra]]. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.
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| ==See also==
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| * [[Binary relation]]
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| * [[Relation algebra]]
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| * [[Demonic composition]]
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| * [[Function composition]]
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| * [[Join (SQL)]]
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| * [[Logical matrix]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.
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| [[Category:Mathematical relations]]
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Nice to meet you, my title is Ling and I totally dig that title. Delaware is the location I love most but I require to move for my family members. Bookkeeping is what I do for a living. Playing crochet is a factor that I'm totally addicted to.
Feel free to visit my web blog: http://Ghaziabadmart.com