Definition:Direct Image Mapping/Relation

Definition
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

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


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

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

Note that:
 * $f_\mathcal R \left({S}\right) = \operatorname{Im} \left({\mathcal R}\right)$

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

Also defined as
Many authors only bother to define this concept when $\mathcal R$ is itself a mapping, say $g$.

Also denoted as
Some sources use $\mathcal R^\to$ for what denotes as $f_\mathcal R$.

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

Also see

 * Definition:Mapping Induced on Powerset by Mapping


 * Definition:Image of Subset under Relation


 * Mapping Induced on Power Set is Mapping, which proves that $f_\mathcal R$ is indeed a mapping.