# 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 1

The **inverse image mapping** of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$:

- $\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$

### Definition 2

The **inverse image mapping** of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$:

- $f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$:

That is:

- $\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## 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 1975: T.S. Blyth: *Set Theory and Abstract Algebra*.

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

## Also see

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

- Results about
**inverse image mappings**can be found**here**.

### Generalizations