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