Definition:Union Relation

Definition

Let:

$(1): \quad \RR_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$

$(2): \quad \RR_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$

Let $\RR_1$ and $\RR_2$ be combinable, that is, that they agree on $S_1 \cap S_2$.

Then the union relation (or combined relation) $\RR$ of $\RR_1$ and $\RR_2$ is:

$\RR \subseteq \paren {S_1 \cup S_2} \times \paren {T_1 \cup T_2}: \map \RR s =

\begin{cases}

\map {\RR_1} s : & s \in S_1 \\
\map {\RR_2} s : & s \in S_2

\end{cases}$