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 [Definition:Image|Image]] be used as appropriate.

Also see

 * Domain
 * Codomain


 * Image
 * Preimage