Definition:Direct Image Mapping

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

Then $\mathcal R$ defines (or induces) a mapping from the power set of $S$ to the power set of $T$:


 * $f_\mathcal R: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right): f_\mathcal R \left({X}\right) = \mathcal R \left({X}\right)$

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 known as
This is sometimes called the direct image of $X$ under $\mathcal R$.

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

Some authors, for example, use $g^\to$ for what we would call $f_g$.

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

Also see

 * Definition:Image of Subset under Relation


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