Codomain

Revision as of 16:14, 29 June 2006 by JBL (talk | contribs)

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 non-negative integers, but its codomain could be any set which contains the non-negative 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 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.

Invalid username
Login to AoPS