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.
The principle of surjectivity: every point in the target set (Y) is hit at least once.
Graphs of three surjective functions between real intervals
A special case of surjectivity: The target set (Y) consists of only one element.
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.
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