Difference between revisions of "Codomain"
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− | 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 15:14, 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.