Definition:Range of Relation

Definition
Let $\mathcal R \subseteq S \times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation).

The range of $\mathcal R$, denoted is defined as one of two things, depending on the source.

It is usually denoted $\operatorname{Rng} \left({\mathcal R}\right)$ or $\operatorname{Ran} \left({\mathcal R}\right)$ (or the same all in lowercase).

Range as Codomain
The range of a relation $\mathcal R \subseteq S \times T$ can be defined as the set $T$.

As such, it is the same thing as the term codomain of $\mathcal R$.

Range as Image
The range of a relation $\mathcal R \subseteq S \times T$ can also be defined as:
 * $\operatorname{Rng} \left({\mathcal R}\right) = \left\{{t \in T: \exists s \in S: \left({s, t}\right) \in \mathcal R}\right\}$

Defined like this, it is the same as what the image of $\mathcal R$.

Beware
Because of the ambiguity in definition, it is often advised that the term range not be used in this context at all, but instead that the term Codomain or Image be used as appropriate.

Also see

 * Definition:Domain (Set Theory)
 * Definition:Codomain (Set Theory)


 * Definition:Image
 * Definition:Preimage