Definition:Image (Relation Theory)/Relation/Relation

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

The image of $\mathcal R$ is the set:


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

Also known as
The image of $\mathcal R$ is often seen referred to as the image set of $\mathcal R$.

Some sources refer to this as the direct image of a relation, in order to differentiate it from an inverse image.

Rather than apply a relation $\mathcal R$ directly to a subset $A$, those sources prefer to define the mapping induced by $\mathcal R$ as a separate concept in its own right.

Other sources call the image of $\mathcal R$ its range, but this convention is discouraged because of potential confusion.

Also see

 * Definition:Mapping, in which the context of an image is usually encountered.


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


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