Definition:Image (Relation 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 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 Element under Mapping


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


 * Definition:Preimage of Element under Relation (also known as Definition:Inverse Image)