Surjective function

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A surjective function; X is the definition set and Y the target set.

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.


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.


  • 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


Web links

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