# Definition:Image (Set Theory)/Relation

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## Contents

## Definition

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

### Image of a Relation

The **image** of $\mathcal R$ is the set:

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

### Image of an Element

Let $s \in S$.

The **image of $s$ by** (or **under**) **$\mathcal R$** is defined as:

- $\map {\mathcal R} s := \set {t \in T: \tuple {s, t} \in \mathcal R}$

That is, $\map {\mathcal R} s$ is the set of all elements of the codomain of $\mathcal R$ related to $s$ by $\mathcal R$.

### Image of a Subset

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

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

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

## 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 $\mathcal R$ directly to a subset $A$, those sources prefer to define the direct image mapping of $\mathcal R$ as a separate concept in its own right.

## Also see

- Definition:Preimage of Relation (also known as Definition:Inverse Image)

### Special cases

- Definition:Image of Mapping, in which the context of an
**image**is usually encountered.