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 defined as:


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

That is:
 * $\mathcal R \left [{X}\right] := \mathcal R^\to \left ({X}\right)$

where $\mathcal R^\to$ denotes the mapping induced on the power set of $S$ by $\mathcal R$.

Also known as
The image of $X$ under $\mathcal R$ is also seen referred to as the direct image of $X$ under $\mathcal R$.

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.

Also see

 * Definition:Image of Subset under Mapping


 * Image of Subset under Relation equals Union of Images of Elements
 * Image of Domain of Relation is Image Set
 * Image of Singleton under Relation