Definition:Composition of Relations

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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


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 $\mathcal R$.


Sources