Preimage of Union under Relation

Theorem
Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

Let $T_1$ and $T_2$ be subsets of $T$.

Then:
 * $\RR^{-1} \sqbrk {T_1 \cup T_2} = \RR^{-1} \sqbrk {T_1} \cup \RR^{-1} \sqbrk {T_2}$

Proof
We have that $\RR^{-1}$ is a relation

The result follows from Image of Union under Relation.

Also see

 * Preimage of Intersection under Relation
 * Image of Intersection under Relation
 * Image of Union under Relation