\begin{definition}[Definition:Composition of Relations]
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, $T_2$.
\end{definition}