composition of relations

Composition of relations relates the elements in a first set to elements in a third set that are mutually related to a same element in a second set. Informally, it "carries" relationship across a mutual pair.

Formally,
if $R \subseteq X \times Y$ and $S \subseteq Y \times Z$ are two binary relations, then their composition $S \circ R$ is the relation

S \circ R = {(x, z) \in X \times Z : \exists y \in Y such that (x, y) \in R and (y, z) \in S}

If the two relations are functions with $f : X \to Y$ and $g : Y \to Z$ ,
then $g \circ f : X \to Z$ ,
and in the way we might write $y = f (x)$ and $z = g (y)$ ,
$(g \circ f) (x) = g (f (x))$ .