Definition:Direct Image Mapping/Mapping

Definition
Let $S$ and $T$ be sets. Let $g \subseteq S \times T$ be a mapping from $S$ to $T$.

Then $g$ defines (or induces) a mapping $f_g: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ from the power set of $S$ to the power set of $T$:


 * $\forall X \in \mathcal P \left({S}\right): f_g \left({X}\right) = g \left[{X}\right]$

where $g \left[{X}\right]$ is the image of $X$ under $g$.

Note that:
 * $f_g \left({S}\right) = \operatorname{Im} \left({g}\right)$

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

Also denoted as
Some sources use $g^\to$ for what denotes as $f_g$.

Similarly, $g^\gets$ is used for $f_{g^{-1}}$, where $g^{-1}$ is the inverse of $g$.

Also see

 * Definition:Mapping Induced on Powerset by Relation


 * Definition:Image of Subset under Mapping


 * Mapping Induced on Power Set is Mapping, which proves that $f_g$ is indeed a mapping.