Surjective function explained

In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a function, the codomain is the image of the function's domain .^{[1] [2]} It is not required that be unique; the function may map one or more elements of to the same element of .

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki,^{[3] [4]} a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

Definition

A surjective function is a function whose image is equal to its codomain. Equivalently, a function

f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y . Surjections are sometimes denoted by a two-headed rightwards arrow,^[5] as in f\colon X\twoheadrightarrow Y .

Symbolically,

If

f\colon X → Y , then f is said to be surjective if\forall y \in Y, \exists x \in X, f(x) = y .^[6]

Examples

• For any set X, the identity function id_X on X is surjective.
• The function defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1) is surjective.
• The function defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number y, we have an x such that f(x) = y: such an appropriate x is (y − 1)/2.
• The function defined by f(x) = x³ − 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x³ − 3x − y = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not injective (and hence not bijective), since, for example, the pre-image of y = 2 is . (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)
• The function defined by g(x) = x² is not surjective, since there is no real number x such that x² = −1. However, the function defined by (with the restricted codomain) is surjective, since for every y in the nonnegative real codomain Y, there is at least one x in the real domain X such that x² = y.
• The natural logarithm function

Notes and References

1. Web site: Injective, Surjective and Bijective. www.mathsisfun.com. 2019-12-07.
2. Web site: Bijection, Injection, And Surjection Brilliant Math & Science Wiki. brilliant.org. en-us. 2019-12-07.
3. .
4. Book: Mashaal, Maurice. Bourbaki. 2006. American Mathematical Soc.. 978-0-8218-3967-6. 106. en.
5. Web site: Arrows – Unicode. 2013-05-11.
6. Web site: Injections, Surjections, and Bijections. Farlow. S. J.. Stanley Farlow . math.umaine.edu. 2019-12-06.