Image of Union under Mapping

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

Let $f: S \to T$ be a mapping.

Let $A$ and $B$ be subsets of $S$.

Then:
 * $f \left[{A \cup B}\right] = f \left[{A}\right] \cup f \left[{B}\right]$

Proof
As $f$, being a mapping, is also a relation, we can apply Image of Union under Relation:


 * $\mathcal R \left[{A \cup B}\right] = \mathcal R \left[{A}\right] \cup \mathcal R \left[{B}\right]$

Also see

 * Preimage of Union under Mapping
 * Image of Intersection under Mapping
 * Preimage of Intersection under Mapping