Definition:Range of Relation/Codomain

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Definition

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


The range of $\RR$ can be defined as $T$.

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


Warning

Because of the ambiguity in definition, it is advised that the term range not be used in this context at all.

Instead that the term Codomain or Image be used as appropriate.

This is the approach to be taken consistently in $\mathsf{Pr} \infty \mathsf{fWiki}$.


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


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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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