Definition:Direct Image Mapping/Relation

Definition

Let $S$ and $T$ be sets.

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

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

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

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

Direct Image Mapping as Set of Images of Subsets

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

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

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

Also defined as

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

Also known as

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

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 $\powerset {\mathcal R}$; see the contravariant power set functor.

Also see

• Results about direct image mappings can be found here.