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:


 * $\mathcal R \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$

Image of Subset as Element of Direct Image Mapping
The image of $X$ by $\mathcal R$ can be seen to be an element of the codomain of the direct image mapping $\mathcal R^\to: \powerset S \to \powerset T$ of $\mathcal R$:


 * $\forall X \in \powerset S: \map {\mathcal R^\to} X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$

Thus:
 * $\forall X \subseteq S: \mathcal R \sqbrk X = \map {\mathcal R^\to} X$

and so the image of $X$ under $\mathcal R$ is also seen referred to as the direct image of $X$ under $\mathcal R$.

Both approaches to this concept are used in.

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


 * Definition:Preimage of Subset under Relation
 * Definition:Preimage of Subset under Mapping