The Surjection reference article from the English Wikipedia on 24-Jul-2004
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Surjection

See the real Africa
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain.

More formally, a function fX → Y is surjective if, for every y in the codomain Y, there is at least one x in the domain X with f(x) = y. Put another way, f is surjective if its range f(X) is equal to the codomain Y, or equivalently, if every element in the codomain has a preimage.

Image:ontoMap.png
Surjective, not injective
Image:mathmap.png
Injective, not surjective
Image:bijMap.png
Bijective
Image:mathmap2.png
Not surjective, not injective
.

Table of contents
1 Examples and counterexamples
2 Properties
3 See also

Examples and counterexamples

On the other hand, the function gR → R defined by g(x) = x2 is not surjective, because (for example) there is no real number x such that x2 = −1.

However, if we define the function hR → [0, ∞) by the same formula as g, but with the codomain restricted to only the nonnegative real numbers, then the function h is surjective. This is because, given an arbitrary nonnegative real number y, we can solve y = x2 to get solutions x = √y and x = −√y.

Properties

See also

Injective function, Bijection