Definition:Direct Image Mapping/Mapping

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

The mapping $f$ induces a mapping $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ from the power set of $S$ to the power set of $T$:


 * $\forall X \in \mathcal P \left({S}\right): f^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

$f^\to$ is referred to as the mapping induced by $f$ (on the power set of $S$).

Note that:
 * $f^\to \left({S}\right) = \operatorname{Im} \left({f}\right)$

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

Also known as
The mapping induced by $f$ is also referred to as the mapping defined by $f$.

Some sources refer to this as the direct image mapping to distinguish it from its inverse.

Also denoted as
The notation used here is that found in.

Many sources use the same notation for the induced mapping as for the mapping itself, but this can cause confusion.

The induced mapping is also denoted $\mathcal P \left({f}\right)$; see the covariant power set functor.

Some sources use $f_g$ to denote what denotes as $g^\to$, but this is confusing and is to be avoided.

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 $f^\to$ is indeed a mapping.


 * Definition:Mapping Induced on Powerset by Inverse of Mapping, where the notation $f^\gets$ is used for the mapping induced by $f^{-1}$.