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Surjection : Surjective

A function f: X → Y is called surjective or onto or a surjection 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, the range f(X) is equal to the codomain Y.

Surjective, not injective
Injective, not surjective

Bijective
Not surjective, not injective

When X and Y are both the real line R, then a surjective function f: R → R can be visualized as one whose graph will be intersected by any horizontal line.

Examples and counterexamples

Consider the function f: R → R defined by f(x) = 2x + 1. This function is surjective, since given an arbitrary real number y, we can solve y = 2x + 1 for x to get a solution x = (y − 1)/2.

On the other hand, 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 we define the function h: R → R⁺ by the same formula as g, but with the codomain has been 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 f: X → Y is surjective if and only if there exists a function g: Y → X such that f ^o g equals the identity function on Y. (This statement is equivalent to the axiom of choice.)
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.
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 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 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 f: X → 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.)