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.

Also known as
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 $f \left({X}\right)$.

Some authors stipulate the name further by calling $\operatorname {Im} \left ({X}\right)$ the direct image of $X$ (by $f$).

This is done to emphasize the distinction between this and the concept of the inverse image.