# Definition:Direct Image Mapping/Mapping

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## Definition

Let $S$ and $T$ be sets.

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

Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.

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

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

## Direct Image Mapping as Set of Images of Subsets

The direct image mapping of $f$ can be seen to be the set of images of all the subsets of the domain of $f$:

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

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

## Also known as

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

The word defined can sometimes be seen instead of induced.

## Also denoted as

The notation used here is that found in 1975: T.S. Blyth: Set Theory and Abstract Algebra.

The direct image mapping is also denoted $\powerset f$; see the covariant power set functor.

## Also see

• Results about direct image mappings can be found here.