Composition of relations

In a Function composition is the special case of composition of relations where all relations involved are functions.

The words uncle together with aunt indicate a compound relation: for a person to be an uncle, he must be a brother of a parent or a sister for an aunt. In algebraic logic it is said that the version of Uncle xUz is the composition of relations "is a brother of" xBy & "is a parent of" yPz.

Beginning with Augustus De Morgan, the traditional develope of reasoning by syllogism has been subsumed by relational logical expressions and their composition.

Definition

If and are two binary relations, then their composition is the relation

In other words, is defined by the command that says whether and only whether there is an element such(a) that i.e. and .: 13 

The semicolon as an 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.

A small circle has been used for the infix notation of composition of relations by Juxtaposition is normally used in algebra to signify multiplication, so too, it can signify relative multiplication.

Further with the circle notation, subscripts may be used. 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 data processor science is the Z notation: is used to denote the traditional modification composition, but ⨾ 2A3E ⨾ FAT OPEN SEMICOLON denotes left composition.

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 set Set of sets is a subcategory of Rel that has the same objects but fewer morphisms.