Definition:Range of Relation/Image

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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:

$\Rng \RR = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$

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


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 $\map {\mathrm {Ran} } \RR$ for the range of a relation (or the same all in lowercase).

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

Some use $R_\RR$.

Also see