Definition:Inverse Image Mapping/Mapping

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

Let $\mathcal P \left({T}\right)$ and $\mathcal P \left({S}\right)$ be their power sets. Let $f: S \to T$ be a mapping.

Definition 1
The inverse image mapping of $f$ is the mapping $f^\gets: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$ that sends a subset $X \subset S$ to its preimage $f^{-1}(X)$ under $f$.

Definition 2
The inverse image mapping of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$:
 * $f^\gets= \left({f^{-1} }\right)^\to: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$
 * $\forall X \in \mathcal P \left({T}\right): f^\gets \left({X}\right) = \left\{ {s \in S: \exists s \in X: \left({t, s}\right) \in f^{-1}}\right\}$

Note that:
 * $f^\gets \left({T}\right) = \operatorname{Im}^{-1} \left({f}\right)$

where $\operatorname{Im}^{-1} \left({f}\right)$ is the preimage of $f$.

Also defined as
Many authors define this concept only when $f$ is itself a mapping.

Also known as
This inverse image mapping of $f$ is also known as the induced mapping on power sets by the inverse $f^{-1}$.

Also denoted as
The notation used here is found in.

The inverse image mapping can also be denoted $\overline{\mathcal P}(f)$; see the contravariant power set functor.

Also see

 * Mapping Induced on Power Set is Mapping, which proves that $\mathcal R^\to$ is indeed a mapping for any relation $\mathcal R$. As $f^{-1}$ is itself a relation, this also holds for $f^\gets = \left({f^{-1} }\right)^\to$.
 * Definition:Direct Image Mapping
 * Definition:Power Set Functor

Generalizations

 * Definition:Inverse Image Mapping of Relation