# Definition:Image (Set Theory)/Relation/Element

## Definition

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

Let $s \in S$.

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

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

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

Thus:

- $\mathcal R \paren s$

is another way to write:

- $\mathcal R \left [{\set s}\right]$

where $\mathcal R \left [{\set s}\right]$ denotes the image of a subset of $\mathcal R$.

## Also denoted as

The symbol $\operatorname {Im}$ can also be used as follows:

For $s \in S$:

- $\operatorname {Im}_\mathcal R \left ({s}\right) := \mathcal R \left [{s}\right]$

but this notation is rarely seen.

Some sources use $\mathcal R \left [{s}\right]$ instead of $\mathcal R \paren s$, but it is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ to keep the notations for the **image** of an element separate from that for the image of a subset.

## Warning

The two notations $s \mathrel {\mathcal R} t$ and $\mathcal R \paren s = t$ do *not* mean the same thing.

The first means:

**$s$ is related to $t$ by $\mathcal R$**

which does not exclude the possibility of there being other elements of $T$ to which $s$ relates.

The second means:

**The complete set of elements of $T$ to which $s$ relates consists of $\set t$**.

## Also see

- Definition:Image of Subset under Relation
- Definition:Domain of Relation
- Definition:Codomain of Relation
- Definition:Range

## Sources

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.11$: Relations: Definition $11.3$