Definition:Direct Image Mapping/Relation

Definition
Let $S$ and $T$ be sets.

Let $\mathcal P(S)$ and $\mathcal P(T)$ 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 $\mathcal R^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ that sends a subset $X\subseteq T$ to its image under $\mathcal R$:
 * $\forall X \in \mathcal P \left({S}\right): \mathcal R^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}\right\}$

Note that:
 * $\mathcal R^\to \left({S}\right) = \operatorname{Im} \left({\mathcal R}\right)$

where $\operatorname{Im} \left({\mathcal R}\right)$ 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