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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
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