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.

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.

The direct image mapping can also be denoted $\powerset {\mathcal R}$; see the contravariant power set functor.

Also see

 * Direct Image Mapping of Relation is Mapping, which proves that $\mathcal R^\to$ is indeed a mapping.


 * Direct Image Mapping of Domain is Image Set of Relation


 * Definition:Image of Subset under Relation


 * Preimage of Subset under Relation equals Union of Preimages of Elements

Special Cases

 * Definition:Direct Image Mapping of Mapping

Generalizations

 * Definition:Image of Relation

Related Concepts

 * Definition:Preimage of Subset under Relation
 * Definition:Preimage of Subset under Mapping


 * Definition:Inverse Image Mapping of Relation
 * Definition:Inverse Image Mapping of Mapping