# Difference between revisions of "Codomain"

ComplexZeta (talk | contribs) m |
|||

Line 1: | Line 1: | ||

− | Let <math>A</math> and <math>B</math> be any [[set]]s, and let <math>f:A\to B</math> be a function. Then <math>B</math> is said to be the '''codomain''' of <math>f</math>. | + | Let <math>A</math> and <math>B</math> be any [[set]]s, and let <math>f:A\to B</math> be a [[function]]. Then <math>B</math> is said to be the '''codomain''' of <math>f</math>. |

+ | In general, a function given by a fixed rule on a fixed domain may have many different codomains. For instance, consider the function <math> f </math> given by the rule <math> f(x) = x^2 </math> whose domain is the integers. The [[range]] of this function is the non-negative integers, but its codomain could be any set which contains the non-negative integers, such as the integers (<math>f:\mathbb{Z}\to\mathbb{Z}</math>), the rationals (<math>f:\mathbb{Z}\to\mathbb{Q}</math>), the reals (<math>f:\mathbb{Z}\to\mathbb{R}</math>), the complex numbers (<math>f:\mathbb{Z}\to\mathbb{C}</math>), or the set <math>\mathbb{Z}_{\geq 0} \cup \{\textrm{Groucho, Harpo, Chico}\}</math>. In this last case, there are exactly three elements of the codomain which are not in the range. Technically, each of these is a different function. (Of course, a function given by the same rule could also take a variety of different domains as well.) | ||

+ | |||

+ | A function is [[surjection|surjective]] exactly when the range is equal to the codomain. | ||

{{stub}} | {{stub}} |

## Revision as of 16:14, 29 June 2006

Let and be any sets, and let be a function. Then is said to be the **codomain** of .

In general, a function given by a fixed rule on a fixed domain may have many different codomains. For instance, consider the function given by the rule whose domain is the integers. The range of this function is the non-negative integers, but its codomain could be any set which contains the non-negative integers, such as the integers (), the rationals (), the reals (), the complex numbers (), or the set . In this last case, there are exactly three elements of the codomain which are not in the range. Technically, each of these is a different function. (Of course, a function given by the same rule could also take a variety of different domains as well.)

A function is surjective exactly when the range is equal to the codomain.
*This article is a stub. Help us out by expanding it.*