# Definition:Union Relation

## Definition

Let:

$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$
$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$

Let $\mathcal R_1$ and $\mathcal R_2$ be combinable, that is, that they agree on $S_1 \cap S_2$.

Then the union relation (or combined relation) $\mathcal R$ of $\mathcal R_1$ and $\mathcal R_2$ is:

$\mathcal R \subseteq \left({S_1 \cup S_2}\right) \times \left({T_1 \cup T_2}\right): \mathcal R \left({s}\right) = \begin{cases} \mathcal R_1 \left({s}\right) : & s \in S_1 \\ \mathcal R_2 \left({s}\right) : & s \in S_2 \end{cases}$