Definition:Direct Image Mapping/Mapping

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.

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.

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

Also see

 * Direct Image Mapping of Mapping is Mapping, which proves that $f^\to$ is indeed a mapping.
 * Definition:Inverse Image Mapping, where the notation $f^\gets$ is used for the mapping induced by $f^{-1}$.


 * Direct Image Mapping of Domain is Image Set of Mapping


 * Definition:Image of Subset under Mapping

Generalizations

 * Definition:Direct Image Mapping of Relation
 * Definition:Image of Mapping

Related Concepts

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


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