Image of Union under Relation

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

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

Let $S_1$ and $S_2$ be subsets of $S$.

Then:
 * $\mathcal R \left({S_1 \cup S_2}\right) = \mathcal R \left({S_1}\right) \cup \mathcal R \left({S_2}\right)$

That is, the image of the union of subsets of $S$ is equal to the union of their images.

Also see

 * Image of Intersection
 * Preimage of Intersection
 * Preimage of Union