Definition:Image (Relation Theory)/Relation/Relation

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

The image of $\RR$ is the set:


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

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

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 $\RR$ directly to a subset $A$, those sources often prefer to define the direct image mapping of $\RR$ as a separate concept in its own right.

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

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

Hence on it is preferred that $\Img \RR$ 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)