# 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:

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

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

Thus:

$\map {\mathcal R} s$

is another way to write:

$\mathcal R \sqbrk {\set s}$

where $\mathcal R \sqbrk {\set s}$ denotes the image of a subset of $\mathcal R$.

## Also denoted as

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

For $s \in S$:

$\map {\operatorname {Img}_\mathcal R} s := \mathcal R \sqbrk s$

but this notation is rarely seen.

Some sources use $\mathcal R \sqbrk s$ instead of $\map {\mathcal R} s$, 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 $\map {\mathcal R} s = t$ 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 $\set t$.