Codomain

Given a function <math>f\colon A\rightarrow B<math>, 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:

<math>f\colon \mathbb{R}\rightarrow\mathbb{R}<math>

defined by

<math>f\colon\,x\mapsto x^2.<math>

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,∞):

<math>0\leq f(x)<\infty.<math>

One could have defined the function g thus:

<math>g\colon\mathbb{R}\rightarrow\mathbb{R}^+<math>

<math>g\colon\,x\mapsto x^2.<math>

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.

See also

• Domain (mathematics)
• Range (mathematics)