Definition:Composition of Relations

Definition
Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be relations.

Then the composite of $\mathcal R_1$ and $\mathcal R_2$ is defined and denoted as:


 * $\mathcal R_2 \circ \mathcal R_1 := \left\{{\left({x, z}\right) \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \left({x, y}\right) \in \mathcal R_1 \land \left({y, z}\right) \in \mathcal R_2}\right\}$

Some authors write $\mathcal R_2 \circ \mathcal R_1$ as $\mathcal R_2 \mathcal R_1$.

It is clear that the composite relation $\mathcal R_2 \circ \mathcal R_1$ can also be defined as:


 * $\mathcal R_2 \circ \mathcal R_1 \left({S_1}\right) = \mathcal R_2 \left({\mathcal R_1 \left({S_1}\right)}\right)$

Note that:
 * $(1): \quad \mathcal R_2 \circ \mathcal R_1 \subseteq S_1 \times T_2$
 * $(2): \quad$ The domain of $R_2 \circ \mathcal R_1$ equals the domain of $\mathcal R_1$, that is, $S_1$
 * $(3): \quad$ The codomain of $R_2 \circ \mathcal R_1$ equals the codomain of $\mathcal R_2$, that is, $T_2$.

Also see

 * Image and Preimage of Composite Relation


 * Composition of Mappings is Composition of Relations

Illustration
The following is an Euler diagram illustrating the relations between the various entities.


 * CompositionOfRelations.png

In the above:
 * $\operatorname{Im} \left({\mathcal R}\right)$ denotes the image of a relation $\mathcal R$
 * $\operatorname{Im}^{-1} \left({\mathcal R}\right)$ denotes the preimage of a relation $\mathcal R$.