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Concept
Composition of relations
Résumé
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation R; S from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions.
The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle () is the composition of relations "is a brother of" () and "is a parent of" ().
Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition.
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 (that is, and ).
The semicolon as an infix notation for composition of relations dates back to Ernst Schroder's textbook of 1895. Gunther Schmidt has renewed the use of the semicolon, particularly in Relational Mathematics (2011). The use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in , as well as the notation for dynamic conjunction within linguistic dynamic semantics.
A small circle has been used for the infix notation of composition of relations by John M. Howie in his books considering semigroups of relations. However, the small circle is widely used to represent composition of functions which reverses the text sequence from the operation sequence. The small circle was used in the introductory pages of Graphs and Relations until it was dropped in favor of juxtaposition (no infix notation). Juxtaposition is commonly used in algebra to signify multiplication, so too, it can signify relative multiplication.
Further with the circle notation, subscripts may be used.
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