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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.

See also

Science Fair Project Encyclopedia Contents Page
09-23-2007 01:00:40
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