Definition:Range of Relation

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This page is about Range of Relation. For other uses, see Range.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ it is denoted $\Rng \RR$, but this may be 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 $\map {\mathrm {Ran} } \RR$ (or the same all in lowercase).

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

Some use $R_\RR$.


Also see


Sources

Those that define $\Rng \RR$ as image:


Those that define $\Rng \RR$ as codomain:


Those which do not use the term at all, but use codomain and image instead:

also offering up range as a synonym for both codomain and image


Some sources brush the question aside by refraining from giving a name to this concept at all:

A map or function (the terms are used interchangeably) between sets $A, B$ is written $f: A \to B$.
We call $A$ the domain of $f$, and we avoid calling $B$ anything.
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