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 R ∘ S what is defined here to be S ∘ R. 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 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 semicolon with Unicode code point U+2A3E) denotes left composition. This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory.
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.
Read more about this topic: Composition Of Relations
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