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 | + | 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 [[integer]]s. The [[range]] of this function is the [[nonnegative]] integers, but its codomain could be any set which contains the nonnegative integers, such as the integers (<math>f:\mathbb{Z}\to\mathbb{Z}</math>), the [[rational]]s (<math>f:\mathbb{Z}\to\mathbb{Q}</math>), the [[real]]s (<math>f:\mathbb{Z}\to\mathbb{R}</math>), the [[complex number]]s (<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 examples 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. | A function is [[surjection|surjective]] exactly when the range is equal to the codomain. | ||
Latest revision as of 10:54, 29 January 2007
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 nonnegative integers, but its codomain could be any set which contains the nonnegative 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 examples 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.
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