Definition:Union Relation

Let:


 * $$\mathcal{R}_1 \subseteq S_1 \times T_1$$ be a relation on $$S_1 \times T_1$$;
 * $$\mathcal{R}_2 \subseteq S_2 \times T_2$$ be a relation on $$S_2 \times T_2$$;
 * $$X = S_1 \cap S_2$$.

Let $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ be combinable, that is, that they agree on $$X$$.

Then the union relation (or combined relation) $$\mathcal{R}$$ of $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ is:

$$\mathcal{R} \subseteq \left({S_1 \cup S_2}\right) \times \left({T_1 \cup T_2}\right): \mathcal{R} \left({s}\right) = \begin{cases} \mathcal{R}_1 \left({s}\right) : & s \in S_1 \\ \mathcal{R}_2 \left({s}\right) : & s \in S_2 \end{cases} $$