Surjective function

In mathematics, a surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function $f$ such that, for every element $y$ of the function's codomain, there exists at least one element $x$ in the function's domain such that $f (x) = y$ . In other words, for a function $f : X \to Y$ , the codomain $Y$ is the image of the function's domain $X$ .^[1]^[2] It is not required that $x$ be unique; the function $f$ may map one or more elements of $X$ to the same element of $Y$ .

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.

^ "Injective, Surjective and Bijective". www.mathsisfun.com. Retrieved 2019-12-07.
^ "Bijection, Injection, And Surjection | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-07.
^ Miller, Jeff, "Injection, Surjection and Bijection", Earliest Uses of Some of the Words of Mathematics, Tripod.
^ Mashaal, Maurice (2006). Bourbaki. American Mathematical Soc. p. 106. ISBN 978-0-8218-3967-6.

[:0-1] "Injective, Surjective and Bijective". www.mathsisfun.com. Retrieved 2019-12-07.

[:1-2] "Bijection, Injection, And Surjection | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-07.

[3] Miller, Jeff, "Injection, Surjection and Bijection", Earliest Uses of Some of the Words of Mathematics, Tripod.

[4] Mashaal, Maurice (2006). Bourbaki. American Mathematical Soc. p. 106. ISBN 978-0-8218-3967-6.

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Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
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