Definition:Inverse Image Mapping/Mapping

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

Let $f^{-1}: T \to S$ be the inverse of $f$.

Let $\left({f^{-1} }\right)^\to: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right)$ be the mapping induced by $f^{-1}$ on the power set of $T$:


 * $\forall X \in \mathcal P \left({T}\right): \left({f^{-1} }\right)^\to \left({X}\right) = \left\{ {s \in S: \exists s \in X: \left({t, s}\right) \in f^{-1}}\right\}$

$\left({f^{-1} }\right)^\to$ is denoted $f^\gets$, and is referred to as the mapping induced by the inverse of $f$ (on the power set of $T$).

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 denoted as
The notation used here is found in.

Also see

 * Definition:Mapping Induced on Powerset by Inverse Relation


 * Definition:Preimage of Subset under Mapping


 * 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 $f^\gets = \left({f^{-1} }\right)^\to$.