Composition of relations

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,^[1] and its result is called a relative product.^[2]: 40 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 (${\displaystyle xUz}$) is the composition of relations "is a brother of" (${\displaystyle xBy}$) and "is a parent of" (${\displaystyle yPz}$). $${\displaystyle U=BP\quad {\text{ is equivalent to: }}\quad xUz{\text{ if and only if }}\exists y\ xByPz.}$$

Beginning with Augustus De Morgan,^[3] the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition.^[4]