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 $\mathcal R$:
 * $\forall X \in \powerset S: \map {\mathcal R^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$

Note that:
 * $\map {\mathcal R^\to} S = \Img {\mathcal R}$

where $\Img {\mathcal R}$ is the image set of $\mathcal R$.

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

Also known as
The direct image mapping of $\mathcal R$ is also known as the mapping induced on power sets by $\mathcal R$ or the mapping defined by $\mathcal R$.

Also denoted as
The notation used here is derived from similar notation for the mapping induced by a mapping found in.

Also see

 * Mapping Induced on Power Set is Mapping, which proves that $\mathcal R^\to$ is indeed a mapping.

Special cases

 * Definition:Direct Image Mapping of Mapping