Definition:Range of Relation

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

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

On it is denoted $\Rng \RR$, but this may non-standard.

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

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

Range as Image
The range of a relation $\RR \subseteq S \times T$ can also be defined as:
 * $\Rng \RR = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$

Defined like this, it is the same as what is defined as the image set of $\RR$.

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 denoted as
Some sources use the notation $\map {\operatorname {Ran} } \RR$ (or the same all in lowercase).

Some sources use $\map {\mathsf {Ran} } \RR$.

Also see

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