Definition:Image (Relation Theory)/Relation/Subset

Definition
Let $\mathcal R \subseteq S \times T$ be a relation. Let $A \subseteq S$ be a subset of $S$.

Then the image set (of $A$ by $\mathcal R$) is:


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

If $A = \operatorname{Dom} \left({\mathcal R}\right)$, we have:


 * $\operatorname{Im} \left ({\operatorname{Dom} \left({\mathcal R}\right)}\right) = \mathcal R \left ({\operatorname{Dom} \left({\mathcal R}\right)}\right) = \operatorname{Im} \left ({\mathcal R}\right)$

It is also clear that $\forall s \in S: \mathcal R \left ({s}\right) = \mathcal R \left ({\left\{{s}\right\}}\right)$.

While the use of $\operatorname{Im} \left ({A}\right)$ etc. can be useful, it is arguably preferable in some situations to use $\mathcal R \left ({A}\right)$, as this makes it more apparent to exactly what relation the image refers.

Some authors use $\mathcal R^\to \left ({A}\right)$ for what we have here as $\mathcal R \left ({A}\right)$.