Definition:Codomain (Set Theory)/Relation

Definition

The codomain of a relation $\mathcal R \subseteq S \times T$ is the set $T$.

It can be denoted $\Cdm {\mathcal R}$.

A note on terminology

Some sources refer to the codomain of a relation as its range.

However, other sources equate the term range with the image set.

As there exists significant ambiguity as to whether the range is to mean the codomain or image set, it is advised that the term range is not used.

The notation $\Cdm {\mathcal R}$ has not actually been found by this author anywhere in the literature. In fact, outside the field of category theory, no symbol for the concept of codomain has been found, despite extensive searching.

However, using $\operatorname{Cdm}$ to mean codomain is a useful enough shorthand to be worth coining. That is the approach which has been taken on this website.

Also known as

The term codomain is seen in some sources (older, and those coming from the direction of logic) as converse domain, from which the more modern term codomain evolved.

Some sources write codomain as co-domain.

Also defined as

Some sources define the codomain as the image. This non-standard usage is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ because of the source of ambiguity.