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Function codomain

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Given a function fA → 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: RR

defined by

f: xx2

The codomain of f is R, but clearly f(x) never takes negative values. The range is in fact the set R+ of non-negative reals, the interval [0,∞):

0 ≤ f(x) < ∞

One could have defined the function g thus:

g: RR+
g: xx2

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



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