Definition:Inverse Image Mapping/Relation

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

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$.

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


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

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

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

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

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

Also denoted as
The notation used here is derived from similar notation for the mapping induced by the inverse of a mapping found in.

Also see

 * Definition:Mapping Induced on Powerset by Inverse of Mapping


 * Definition:Preimage of Subset under Relation


 * Mapping Induced on Power Set is Mapping, which proves that $\mathcal R^\to$, and so therefore $\mathcal R^\gets = \left({\mathcal R^{-1} }\right)^\to$, is indeed a mapping for any relation $\mathcal R$.