Surjective function

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In mathematics, a function is said to be surjective or onto if its image is equal to its codomain. A function f: XY is surjective if and only if for every y in the codomain Y there is at least one x in the domain X such that f(x) = y. A surjective function is called a surjection. Surjections are sometimes denoted by a two-headed rightwards arrow, as in f: XY.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,[1] a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

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Examples

For any set X, the identity function idX on X is surjective.

The function fZ → {0,1} defined by f(n) = n mod 2 and mapping even integers to 0 and odd integers to 1 is surjective.

The function fR → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number y we have an x such that f(x) = y: an appropriate x is (y − 1)/2.

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