# Definition:Image (Relation Theory)/Relation/Subset

## Definition

Let $\RR \subseteq S \times T$ be a relation.

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

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

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

### Image of Subset as Element of Direct Image Mapping

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

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

Thus:

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

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

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