Composition of relations

In mathematics, the composition of binary relations is a concept of forming a new relation S ∘ R from two given relations R and S, having as its best-known special case the composition of functions.

If ${\displaystyle R\subseteq X\times Y}$ and ${\displaystyle S\subseteq Y\times Z}$ are two binary relations, then their composition ${\displaystyle S\circ R}$ is the relation

In other words, ${\displaystyle S\circ R\subseteq X\times Z}$ is defined by the rule that says ${\displaystyle (x,z)\in S\circ R}$ if and only if there is an element ${\displaystyle y\in Y}$ such that ${\displaystyle x\,R\,y\,S\,z}$ (i.e. ${\displaystyle (x,y)\in R}$ and ${\displaystyle (y,z)\in S}$).

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