Definition:Inverse Image Mapping/Relation

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

Let $\mathcal P \left({T}\right)$ and $\mathcal P \left({S}\right)$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

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

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

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

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

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

Also see

 * 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 $\mathcal R$.

Special cases

 * Definition:Mapping Induced on Powerset by Inverse of Mapping