Image of Intersection under Mapping

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

Let $f: S \to T$ be a mapping. Let $S_1$ and $S_2$ be subsets of $S$.

Then:
 * $f \left[{S_1 \cap S_2}\right] \subseteq f \left[{S_1}\right] \cap f \left[{S_2}\right]$

That is, the image of the intersection of subsets of a mapping is a subset of the intersection of their images.

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


 * $\mathcal R \left[{S_1 \cap S_2}\right] \subseteq \mathcal R \left[{S_1}\right] \cap \mathcal R \left[{S_2}\right]$

Note
Note that equality does not hold in general.

Let:
 * $S_1 = \left\{{x \in \Z: x \le 0}\right\}$
 * $S_2 = \left\{{x \in \Z: x \ge 0}\right\}$
 * $f: \Z \to \Z: \forall x \in \Z: f \left({x}\right) = x^2$

We have:
 * $f \left[{S_1}\right] = \left\{{0, 1, 4, 9, 16, \ldots}\right\} = f \left[{S_2}\right]$

Then:
 * $f \left[{S_1}\right] \cap f \left[{S_2}\right] = \left\{{0, 1, 4, 9, 16, \ldots}\right\}$

but:
 * $f \left[{S_1 \cap S_2}\right] = f \left[{\left\{{0}\right\}}\right] = \left\{{0}\right\}$

Note that from Image of Intersection under Injection equality always holds iff $f$ is an injection.

Also see

 * Image of Intersection under One-to-Many Relation
 * Image of Intersection under Injection


 * Mapping Preimage of Intersection


 * Mapping Image of Union
 * Mapping Preimage of Union