# Definition:Codomain (Set Theory)/Mapping

## Definition

Let $f: S \to T$ be a mapping.

The codomain of $f$ is the set $T$.

It can be denoted $\operatorname{Cdm} \left({f}\right)$ or $\operatorname{Cod} \left({f}\right)$.

## Also known as

Some sources, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra, also refer to the codomain as the arrival set.

On rare occasions, the codomain is referred to as the target.

Some sources write codomain as co-domain.

## A note on terminology

Some sources refer to the codomain of a 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[1].

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 $\operatorname{Cdm} \left({f}\right)$ has not actually been found by this author anywhere in the literature. In fact, except in 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.

## Notes and References

1. 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces: Notation and Terminology:
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.