# Definition:Direct 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 $S \times T$.

The **direct image mapping** of $\RR$ is the mapping $\RR^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq T$ to its image under $\RR$:

- $\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} & : X \ne \O \\ \O & : X = \O \end {cases}$

## Direct Image Mapping as Set of Images of Subsets

The **direct image mapping** of $\RR$ can be seen to be the set of images of all the subsets of the domain of $\RR$.

- $\forall X \subseteq S: \RR \sqbrk X = \map {\RR^\to} X$

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

Some sources refer to this as the **mapping induced (on the power set) by $\RR$**.

The word **defined** can sometimes be seen instead of **induced**.

## Also denoted as

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

The **direct image mapping** can also be denoted $\powerset \RR$; see the contravariant power set functor.

## Also see

- Direct Image Mapping of Relation is Mapping, which proves that $\RR^\to$ is indeed a mapping.

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

### Special Cases

### Generalizations

### Related Concepts

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations