Definition:Codomain (Relation Theory)

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

It can be denoted $\operatorname{Cdm} \left({\mathcal R}\right)$.

Mapping
The term codomain is usually seen when the relation in question is actually a mapping.

Some sources, for example, also refer to the codomain as the arrival set.

Morphism
If $f : X \to Y$ is a morphism, then the codomain of $f$ is defined to be the object $Y$, often written $Y = \operatorname{cod}f$.

A note on terminology
Some sources refer to the codomain of a relation or mapping as its range.

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

Other sources brush the question aside by refraining from giving the codomain a name at all.

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 $\operatorname{Cdm} \left({\mathcal R}\right)$, or $\operatorname{Cdm} \left({f}\right)$ where $f$ is a mapping, has not actually been found by this author anywhere in the literature. In fact, 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 see

 * Domain
 * Range


 * Image
 * Preimage