Definition:Direct Image Mapping

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Definition

Let $S$ and $T$ be sets.

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

Relation

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 $\mathcal R$:

$\forall X \in \mathcal P \left({S}\right): \mathcal R^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}\right\}$


Mapping

Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.


The direct image mapping of $f$ is the mapping $f^\to: \mathcal P \left({S}\right) \to \mathcal P \left({T}\right)$ that sends a subset $X \subseteq S$ to its image under $f$:

$\forall X \in \mathcal P \left({S}\right): f^\to \left({X}\right) = \left\{ {t \in T: \exists s \in X: f \left({s}\right) = t}\right\}$


Also see

  • Results about direct image mappings can be found here.