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Encyclopedia > Surjective

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.

 Bijective (injective and surjective) Injective, not surjective Surjective, not injective Not surjective, not injective

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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

• A function fX → Y is surjective if and only if there exists a function gY → X such that f o g equals the identity function on Y. (This statement is equivalent to the axiom of choice.)
• By definition, a function is bijective if and only if it is both surjective and injective.
• If f o g is surjective, then f is surjective.
• If f and g are both surjective, then f o g is surjective.
• fX → 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 fX → 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 hX → Z can be decomposed as h = g o f for a suitable surjection f and injection 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.
• If fX → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (This statement is also equivalent to the axiom of choice.)
• 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.

Results from FactBites:

 Surjection - Wikipedia, the free encyclopedia (522 words) In mathematics, a function f is said to be surjective if and only 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. 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. In the language of category theory, surjective functions are precisely the epimorphisms in the category of sets.
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