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.

Inverse Image Mapping as Set of Preimages of Subsets
The inverse image mapping of $f$ can be seen to be the set of preimages of all the subsets of the codomain of $f$.


 * $\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$

Both approaches to this concept are used in.

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

Also known as
The inverse image mapping of $f$ is also known as the preimage mapping of $f$.

Some sources refer to this as the mapping induced (on the power set) by the inverse $f^{-1}$.

Also denoted as
The notation used here is found in.

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

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Mapping


 * Inverse Image Mapping of Codomain is Preimage Set of Mapping


 * Inverse Image Mapping of Mapping is Mapping, which proves that $f^\gets$ is indeed a mapping.


 * Definition:Power Set Functor


 * Definition:Preimage of Subset under Mapping

Generalizations

 * Definition:Inverse Image Mapping of Relation
 * Definition:Preimage of Mapping

Related Concepts

 * Definition:Image of Subset under Mapping
 * Definition:Image of Subset under Relation


 * Definition:Direct Image Mapping of Mapping
 * Definition:Direct Image Mapping of Relation