Codomain

Given a function $f\colon A\rightarrow B$ , 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:

$f\colon \mathbb{R}\rightarrow\mathbb{R}$

defined by

$f\colon\,x\mapsto x^2.$

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, i.e. the interval [0,∞):

$0\leq f(x)<\infty.$

One could have defined the function g thus:

$g\colon\mathbb{R}\rightarrow\mathbb{R}^+$

$g\colon\,x\mapsto x^2.$

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.

Codomain

Example

See also