Codomain
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.