# 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

• Results about direct image mappings can be found here.