Image of Subset under Mapping equals Union of Images of Elements

Theorem

Let $S$ and $T$ be sets.

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

Let $X \subseteq S$ be a subset of $S$.

Then:

$\ds f \sqbrk X = \bigcup_{x \mathop \in X} \map f x$

where:

$f \sqbrk X$ is the image of the subset $X$ under $f$

$\map f x$ is the image of the element $x$ under $f$.

Proof

By definition, a mapping is a relation.

Thus Image of Subset under Relation equals Union of Images of Elements applies.

$\blacksquare$

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Remark $10.8 \ \text{(d)}$