Definition:Direct Image Mapping/Mapping

Definition
Let $S$ and $T$ be sets.

Let $\mathcal P(S)$ and $\mathcal P(T)$ be their power sets. Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.

The mapping induced on power sets by $f$  is the mapping $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ that sends a subset $X \subseteq S$ to its image under $f$:
 * $\forall X \in \mathcal P \left({S}\right): f^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$

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 the inverse image mapping.

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 $f^{-1}$.