# Definition:Codomain (Relation Theory)/Mapping

## Definition

Let $S$ and $T$ be sets.

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

The **codomain** of $f$ is $T$.

It is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\Cdm f$.

## Also known as

The **codomain** of a mapping is sometimes called 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.

For example, from 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.*

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 $\Cdm f$ 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 $\Cdm f$ to mean **codomain** is a useful enough shorthand to be worth coining.

This is the approach which has been taken on this website.

## Also see

## Technical Note

The $\LaTeX$ code for \(\Cdm {X}\) is `\Cdm {X}`

.

When the argument is a single character, it is usual to omit the braces:

`\Cdm X`

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(e)}$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.3$: Functions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**codomain** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**function (map, mapping)** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.1$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**codomain** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**function (map, mapping)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**codomain**

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