Definition:Inverse Image Mapping/Mapping

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

Let $\powerset S$ and $\powerset T$ be their power sets. Let $f: S \to T$ be a mapping.

Definition 2
Note that:
 * $f^\gets \paren T = \Preimg f$

where $\Preimg f$ 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 preimage mapping or 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 $\operatorname {\overline {\mathcal P} } \paren f$; see the contravariant power set functor.

Also see

 * Equivalence of Definitions of Inverse Image Mapping of 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 $f^\gets = \paren {f^{-1} }^\to$.


 * Definition:Direct Image Mapping
 * Definition:Power Set Functor

Generalizations

 * Definition:Inverse Image Mapping of Relation