What is "codomain"

n. (context mathematics English) The target space into which a function maps elements of its domain. It always contains the range of the function, but can be larger than the range if the function is not surjective.

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image.

The codomain is part of a function if it is defined as described in 1954 by Nicolas Bourbaki, namely a triple , with a functional subset of the Cartesian product and is the set of first components of the pairs in (the domain). The set is called the graph of the function. The set of all elements of the form , where ranges over the elements of the domain , is called the image of . In general, the image of a function is a subset of its codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements in its codomain for which the equation does not have a solution.

An alternative definition of function by Bourbaki [Bourbaki, op. cit., p. 77], namely as just a functional graph, does not include a codomain and is also widely used. For example in set theory it is desirable to permit the domain of a function to be a proper class , in which case there is formally no such thing as a triple . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form .