Definition:Composition of Relations

Definition

Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations.

Then the composite of $\RR_1$ and $\RR_2$ is defined and denoted as:

$\RR_2 \circ \RR_1 := \set {\tuple {x, z} \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \tuple {x, y} \in \RR_1 \land \tuple {y, z} \in \RR_2}$

Some authors write $\RR_2 \circ \RR_1$ as $\RR_2 \RR_1$.

It is clear that the composite relation $\RR_2 \circ \RR_1$ can also be defined as:

$\map {\RR_2 \circ \RR_1} {S_1} = \map {\RR_2} {\map {\RR_1} {S_1} }$

Note that:

$(1): \quad \RR_2 \circ \RR_1 \subseteq S_1 \times T_2$

$(2): \quad$ The domain of $R_2 \circ \RR_1$ equals the domain of $\RR_1$, that is, $S_1$

$(3): \quad$ The codomain of $R_2 \circ \RR_1$ equals the codomain of $\RR_2$, that is, $T_2$.

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

In the above:

$\Img \RR$ denotes the image of a relation $\RR$

$\Preimg \RR$ denotes the preimage of a relation $\RR$.