Definition:Image (Set Theory)/Relation/Subset

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Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $X \subseteq S$ be a subset of $S$.


Then the image set (of $X$ by $\mathcal R$) is defined as:

$\mathcal R \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$


Image of Subset as Element of Direct Image Mapping

The image of $X$ by $\mathcal R$ can be seen to be an element of the codomain of the direct image mapping $\mathcal R^\to: \powerset S \to \powerset T$ of $\mathcal R$:

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

Thus:

$\forall X \subseteq S: \mathcal R \sqbrk X = \map {\mathcal R^\to} X$

and so the image of $X$ under $\mathcal R$ is also seen referred to as the direct image of $X$ under $\mathcal R$.


Both approaches to this concept are used in $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see


Special Cases


Generalizations


Related Concepts


Sources