##### Wiktionary

**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.

##### Wikipedia

**Codomain**

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 .