Function codomain

Given a function f: A → B, the set B is called the codomain of f. The codomain is not to be confused with the range f(A), which is in general only a subset of B.

Example

Let the function f be a function on the real numbers:

f: R → R

defined by

f: x → x²

The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R⁺ -- non-negative reals, ie the interval [0,∞):

0 ≤ f(x) < ∞

One could have defined the function g thus:

g: R → R⁺

g: x → x²

While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.

The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not.