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:


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

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.

Many sources denote the image of a relation $\mathcal R$ by $\map {\operatorname {Im} } {\mathcal R}$, but this notation can be confused with the imaginary part of a complex number $\map \Im z$.

Hence on it is preferred that $\Img {\mathcal R}$ be used.

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)