# Definition:Image (Set Theory)/Relation

## Definition

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

### Image of a Relation

The image of $\RR$ is the set:

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

### Image of an Element

Let $s \in S$.

The image of $s$ by (or under) $\RR$ is defined as:

$\map \RR s := \set {t \in T: \tuple {s, t} \in \RR}$

That is, $\map \RR s$ is the set of all elements of the codomain of $\RR$ related to $s$ by $\RR$.

### Image of a Subset

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

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

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

## Notes

Some sources refer to this as the direct image of a relation, in order to differentiate it from an inverse image.

Rather than apply a relation $\RR$ directly to a subset $A$, those sources prefer to define the direct image mapping of $\RR$ as a separate concept in its own right.