# Codomain of Composite Relation

## Theorem

Let $\mathcal R_2 \circ \mathcal R_1$ be a composite relation.

Then the codomain of $\mathcal R_2 \circ \mathcal R_1$ is the codomain of $\mathcal R_2$:

$\Cdm {\mathcal R_2 \circ \mathcal R_1} = \Cdm {\mathcal R_2}$

## Proof

Let $\mathcal R_1 \subseteq S_1 \times S_2$ and $\mathcal R_2 \subseteq S_2 \times S_3$.

The codomain of $\mathcal R_2$ is $S_3$.

The composite of $\mathcal R_1$ and $\mathcal R_2$ is defined as:

$\mathcal R_2 \circ \mathcal R_1 = \set {\tuple {x, z}: x \in S_1, z \in S_3: \exists y \in S_2: \tuple {x, y} \in \mathcal R_1 \land \tuple {y, z} \in \mathcal R_2}$

From this definition:

$\mathcal R_2 \circ \mathcal R_1 \subseteq S_1 \times S_3$

Thus the codomain of $\mathcal R_2 \circ \mathcal R_1$ is $S_3$.

Thus:

$\Cdm {\mathcal R_2 \circ \mathcal R_1} = S_3 = \Cdm {\mathcal R_2}$

$\blacksquare$