Composition of relations

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

Definition
If ''$$R:X\to Y$$ and $$S:Y\to Z$$ are two binary relations, then their compose $$S\circ R$$ is the relation
 * $$S\circ R:X\to Z$$ defined by $$\forall(x,z)\in X\times Z:\exists y\in Y:

x\,R\,y\,S\,z$$ where, as usual, $$ x\,R\,y\,S\,z \iff x\,R\,y\land y\,S\,z $$. In other words, S o R is the relation from X to Z whose graph is
 * $$\operatorname{graph}\,S\circ R = \{ (x,z)\in X\times Z\mid \exists y\in Y: (x,y)\in \operatorname{graph}\,R\land (y,z)\in \operatorname{graph}\,S \}$$.

In particular fields, authors might denote by R o S what is defined here to be S o 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.

Properties
Composition of relations is associative.

The inverse relation of S o R is (S o R)-1 = R-1 o S-1.

The compose of (partial) functions (i.e. functional relations) is again a (partial) function.

If R and S are injective, then S o R is injective, which conversely implies only the injectivity of R.

If R and S are surjective, then S o R is surjective, which conversely implies only the surjectivity of S.

The binary relations on a set X (i.e. relations from X to X) form a monoid for composition, with the identity map on X as neutral element.