Codomain

In mathematics, the codomain or target set of a function is the set $Y$ into which all of the output of the function is constrained to fall. It is the set $Y$ in the notation $f : X \to Y$ . 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 $f$ if it is defined as described in 1954 by Nicolas Bourbaki, namely a triple $(X, Y, F)$ , with $F$ a functional subset of the Cartesian product $X \times Y$ and $X$ is the set of first components of the pairs in $F$ (the domain). The set $F$ is called the graph of the function. 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 $f$ . 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 $y$ in its codomain for which the equation $f (x) = y$ 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 $X$ , in which case there is formally no such thing as a triple $(X, Y, F)$ . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form $f : X \to Y$ .

...
Wikipedia