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>.
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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>.
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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.)
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A function is [[surjection|surjective]] exactly when the range is equal to the codomain.
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Latest revision as of 11:54, 29 January 2007

Let $A$ and $B$ be any sets, and let $f:A\to B$ be a function. Then $B$ is said to be the codomain of $f$.

In general, a function given by a fixed rule on a fixed domain may have many different codomains. For instance, consider the function $f$ given by the rule $f(x) = x^2$ 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 ($f:\mathbb{Z}\to\mathbb{Z}$), the rationals ($f:\mathbb{Z}\to\mathbb{Q}$), the reals ($f:\mathbb{Z}\to\mathbb{R}$), the complex numbers ($f:\mathbb{Z}\to\mathbb{C}$), or the set $\mathbb{Z}_{\geq 0} \cup \{\textrm{Groucho,  Harpo,  Chico}\}$. 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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