Preimage of Union under Mapping

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

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

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

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

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


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

Also see

 * Image of Union under Mapping


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