Codomain

In mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in the notation f : X → Y. The term range is sometimes ambiguously used to refer to either the codomain or the image of a function. A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution. A codomain is not part of a function f if f is defined as just a graph. For example, in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.