# Surjection

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 *f*: *X* → *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.

Surjective, not injective |
Injective, not surjective |

Bijective |
Not surjective, not injective |

Table of contents |

2 Properties 3 See also |

## Examples and counterexamples

However, if we define the function `h`: **R** → [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` = `x`^{2} 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.) - 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. -
`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 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.) - 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.