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:


 * $\operatorname{Im} \left ({s}\right) = \mathcal R \left ({s}\right) = \left\{ {t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

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

If Image is a Singleton
If $\mathcal R \left ({s}\right)$ for some $s \in S$ (or $\mathcal R \left ({S_1}\right)$ for some $S_1 \subseteq S$) has only one element $t \in T$, then we can write:


 * $\mathcal R \left ({s}\right) = t$

instead of:


 * $\mathcal R \left ({s}\right) = \left\{{t}\right\}$

Note: The two notations $s \mathcal R t$ and $\mathcal R \left ({s}\right) = t$ 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 $\left\{ {t}\right\}$.

Also see

 * Definition:Image of Element under Mapping


 * Definition:Domain of Relation
 * Definition:Codomain of Relation
 * Definition:Range


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