Preimage of Union under Relation

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

Let $\mathcal R \subseteq S \times T$ be a relation.

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

Then:
 * $\mathcal R^{-1} \left[{T_1 \cup T_2}\right] = \mathcal R^{-1} \left[{T_1}\right] \cup \mathcal R^{-1} \left[{T_2}\right]$

Proof
This follows from Image of Union under Relation, and the fact that $\mathcal R^{-1}$ is itself a relation, and therefore obeys the same rules.

Also see

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