Definition:Image (Relation Theory)/Mapping/Subset

Definition
Let $f: S \to T$ be a mapping. Let $A \subseteq S$ be a subset of $S$.

Then the image set (of $A$ by $f$) is:


 * $\operatorname{Im} \left ({A}\right) = f \left ({A}\right) = \left\{ {t \in T: \exists s \in A: f \left({s}\right) = t}\right\}$

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


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

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

While the use of $\operatorname{Im} \left ({A}\right)$ etc. can be useful, it is arguably preferable in some situations to use $f \left ({A}\right)$, as this makes it more apparent to exactly what relation the image refers.

Some authors use $f^\to \left ({A}\right)$ for what we have here as $f \left ({A}\right)$.