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 [1]
- 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.
See also: Function domain, Function range
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