Definition:Image (Relation Theory)/Mapping/Subset

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

Let $X \subseteq S$.

Then the image (or image set) of $X$ (by $f$) is defined as:
 * $\operatorname {Im} \left ({X}\right) := \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

If $X = \operatorname{Dom} \left({f}\right)$, we have:


 * $\operatorname{Im} \left ({\operatorname{Dom} \left({f}\right)}\right) = \operatorname{Im} \left ({f}\right)$

where $\operatorname{Im} \left ({f}\right)$ is the image (set) of $f$.

It is also clear that:
 * $\forall s \in S: f \left ({\left\{{s}\right\}}\right) = f \left ({s}\right)$

where $f \left ({s}\right)$ is the image of $s$.

$\operatorname {Im} \left ({X}\right)$ is frequently rendered as $f \left ({X}\right)$, which can be argued as preferable in some situations, as this makes it more apparent to exactly what mapping the image refers.

Some authors prefer not to use the notation $f \left ({X}\right)$ and instead use the concept of the mapping induced from the power set of $S$ to the power set of $T$.

For example:
 * uses $f^\to \left ({X}\right)$ for $f \left({X}\right)$;
 * glosses over the matter, and quietly drops the notation $f \left [{X}\right]$ for $f \left({X}\right)$.