# Definition:Inverse Image Mapping/Relation

## Definition

Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ be their power sets.

Let $\RR \subseteq S \times T$ be a relation on $S \times T$.

### Definition 1

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

- $\forall Y \in \powerset T: \map {\RR^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} } & : \Img \RR \cap Y \ne \O \\ \O & : \Img \RR \cap Y = \O \end {cases}$

### Definition 2

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

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

That is:

- $\forall Y \in \powerset T: \map {\RR^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$

## Inverse Image Mapping as Set of Preimages of Subsets

The **inverse image mapping** of $\RR$ can be seen to be the set of preimages of all the subsets of the codomain of $\RR$.

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

Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also defined as

Many authors define this concept only when $\RR$ is itself a mapping.

## Also known as

This **inverse image mapping** of $\RR$ is also known as the **preimage mapping** of $\RR$.

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

## Also denoted as

The notation used here is derived from similar notation for the **inverse image mapping of a mapping** found in 1975: T.S. Blyth: *Set Theory and Abstract Algebra*.

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

## Also see

- Inverse Image Mapping of Relation is Mapping, which shows that $\RR^\gets$ is indeed a mapping for any relation $\RR$.

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

### Special cases