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In mathematics, the composition of binary relations is a concept of forming a new relation from two given relations ''R'' and ''S'', having as its most well-known special case the composition of functions. == Definition == If and are two binary relations, then their composition is the relation : In other words, is defined by the rule that says if and only if there is an element such that (i.e. and ). In particular fields, authors might denote by what is defined here to be . 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 relations. Some authors〔Kilp, Knauer & Mikhalev, p. 7〕 prefer to write and explicitly when necessary, depending whether the left or the right relation is the first one applied. A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition.〔ISO/IEC 13568:2002(E), p. 23〕〔http://www.fileformat.info/info/unicode/char/2a3e/index.htm〕 This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory,〔http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6〕 as well as the notation for dynamic conjunction within linguistic dynamic semantics.〔http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog〕 The semicolon notation (with this semantic) was introduced by Ernst Schröder in 1895.〔 A free HTML version of the book is available at http://www.cs.man.ac.uk/~pt/Practical_Foundations/〕 The binary relations are sometimes regarded as the morphisms in a category Rel which has the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. The category 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 allegories. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Composition of relations」の詳細全文を読む スポンサード リンク
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