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In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y.
Said another way, a function f: X → Y is surjective if and only if its range f(X) is equal to its codomain Y. A surjective function is called a surjection, and said to be onto.
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Examples and a counterexample
For any set X, the identity function idX on X is surjective.
The function f: R → R defined by f(x) = 2x + 1 is surjective, because for every real number y we have f(x) = y where x is (y - 1)/2.
The function g: R → R defined by g(x) = x² is not surjective, because (for example) there is no real number x such that x² = −1. However, if the codomain is defined as 0,+∞), then g is surjective.
The function f: Z → defined by f(x) = x mod 4 is surjective.
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Properties
If f and g are both surjective, then f o g is surjective.
If f o g is surjective, then f is surjective (but g may not be).
f: X → Y is surjective if and only if, given any functions g,h:Y → Z, whenever g o f = h o f, then g = h. In other words, surjective functions are precisely the epimorphisms in the category '''Set''' of sets.
If f: X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimage f −1(B).
Every function h: X → Z can be decomposed as h = g o f for a suitable surjection f and injective function g. This decomposition is unique up to isomorphism, and f may be thought of as a function with the same values as h but with its codomain restricted to the range h(W) of h, which is only a subset of the codomain Z of h.
By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) A → A/~ be the projection map which sends each x in A to its equivalence class ''x''~, and let fP A/~ → B be the well-defined function given by fP(''x''~) = f(x). Then f = fP o P(~).
If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers.
If both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only if f is injective.
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See also
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Category theory view
In the language of category theory, surjective functions are precisely the epimorphisms in the category of sets.
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