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:
 * $f \left [{X}\right] := \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

Also known as
As well as using the notation $\operatorname{Im} \left ({f}\right)$ to denote the image set of a mapping, the symbol $\operatorname{Im}$ can also be used as follows:

For $X \subseteq S$, we have:
 * $\operatorname{Im}_f \left ({X}\right) := f \left [{X}\right]$

but this notation is rarely seen.

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 $f \left[{X}\right]$

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

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

Also see

 * Image of Singleton under Mapping
 * Image of Domain of Mapping is Image Set
 * Image of Subset under Relation equals Union of Images of Elements