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.
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