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Surjection

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A function fX → 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 fR → R can be visualized as one whose graph will be intersected by any horizontal line.

Examples and counterexamples

Consider the function fR → 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 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 → 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 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.)
  • 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 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.)


See also: Injective function, Bijection



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