# Surjective function

A **surjective function** is a mathematical function that takes each element of the target set *at least* once as a function value. That is, each element of the target set has at least one archetype . A function is always surjective with regard to its number of images .

A surjective function is also known as **surjection** . If it is also injective , it is called bijective . In the language of relations one speaks of **right total functions.**

## definition

Let there be and sets, as well as a mapping.

The mapping is called *surjective* if there is (at least) one from with for each of them . Such a map is also listed as follows: .

Formal: (see existential and universal quantifier ).

## Graphic illustrations

## Examples and counterexamples

- The empty function in a one-element set is probably the simplest example of a nonsurjective function.
- The function with is surjective, because there is a prototype for every real number . From the equation one obtains the equation by means of equivalence conversion with which a prototype can be calculated for each .
- The sine function is surjective. Each horizontal line with cuts the graph of the sine function at least once (even indefinitely).
- However, the sine function is not surjective, since z. B. the straight line has no intersection with the graph, so the value 2 is not assumed as a function value.
- denote the set of complex numbers .

- is not surjective, since z. B. has no archetype.
- is surjective.

## properties

- Note that the surjectivity of a function does not only depend on the function graph but also on the target set (in contrast to injectivity , the presence of which can be seen on the function graph).

- A function is surjective if and only if holds for all .

- If the functions and are surjective, then this also applies to the composition (concatenation)

- It follows from the surjectivity of that is surjective.

- A function is then exactly onto, when a right inverse , has thus a function with (where the identity mapping to hereinafter). This statement is equivalent to the axiom of choice in set theory.

- A function is exactly then surjective if rechtskürzbar is, so if for any functions with already follows. (This property motivates the term epimorphism used in category theory .)

- Any function can be represented as a concatenation , with the function being surjective and injective.

## Cardinalities of sets

For a finite set , the cardinality is simply the number of elements of . If now is a surjective function between finite sets, then at most there can be as many elements as , so it is true

For infinite quantities , the size comparison of thicknesses is defined with the help of the term injection , but here, too, the following applies: If surjective, then the thickness of is not greater than the thickness of here too

## literature

- OA Ivanova: Surjection . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

## Web links

**Wikibooks: Evidence archive: set theory**- learning and teaching materials