Definition:Image (Relation Theory)/Relation/Subset

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

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


 * $\mathcal R \left [{X}\right] = \left\{ {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}\right\}$

If $X = \operatorname{Dom} \left({\mathcal R}\right)$, the domain of $\mathcal R$, we have:


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

where $\operatorname{Im} \left ({\mathcal R}\right)$ is the image of $\mathcal R$.

Also denoted as
As well as using the notation $\operatorname{Im} \left ({\mathcal R}\right)$ to denote the image set of a relation, the symbol $\operatorname{Im}$ can also be used as follows:

For $X \subseteq S$:
 * $\operatorname{Im}_\mathcal R \left ({X}\right) := \mathcal R \left [{X}\right]$

but this notation is rarely seen.

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

Also see

 * Image of Subset under Relation equals Union of Images of Elements
 * Image of Singleton under Relation