Bijection

Bijectivity ( the adjective bijective, which is about, reversible clearly ' means → hence the term one-one -one correspondence or ) is a mathematical term from the field of set theory. He refers to a special feature of mappings and functions. Bijective mappings and functions are also called bijections. To a mathematical structure occurring bijections often have their own names such as isomorphism, diffeomorphism, homeomorphism, mirroring or similar. Here are then usually additional requirements with regard to the conservation of the considered structure to fulfill.

To illustrate, one can say that with a bijection a complete pairing between the elements of set of definitions and target quantity takes place. Bijections treat its domain and its range of values ​​that is symmetrical; therefore a bijective function always has an inverse function.

For a bijection defining the quantity and the target quantity always have the same cardinality. In the event that a bijection between two finite sets present, this common cardinality is a natural number which is exactly the number of elements of each of the two sets.

The bijection of a set onto itself is also called permutation. Again, there are many mathematical structures in their own name. Has the bijection going beyond structure-preserving properties, one speaks of a automorphism.

Definition

Be and quantities; and is a function or mapping that maps from to, ie. is bijective if exactly one exists with for all.

This means is bijective if and only if both

Graphical illustrations

Four bijective strictly increasing real continuous functions.

Four bijective strictly decreasing real continuous functions.

Examples and counter-examples

The set of real numbers is denoted here with the set of non-negative real numbers.

• The function is bijective with the inverse function.
• Also, is bijective for the function with the inverse function.
• Example: Assigns one each ( monogamous ) married people his spouse or his spouse to, this is a bijection of the set of all married people to itself This is actually an example of a self- inverse mapping.

• The following four square functions differ only in their definition or sets of values ​​:

Properties

• Are and finite sets with the same number of elements and is a function, then: Is injective, then it is already bijective. Is surjective, then is already bijective.

• In particular, therefore applies to functions of a finite set into itself:   is injective ⇔ ⇔ surjective is bijective. For infinite sets that is wrong in general. This can be mapped injectively on proper subsets, just as there are surjective images of an infinite set onto itself that are not bijections. Such surprises are described in the article Hilbert's Hotel detail, see also Dedekind infinity.
• The bijective functions, this applies also to the chaining. The inverse function of is then.

• Is bijective, then it is injective and surjective.

• Is a function, and there is a feature that the two equations   ( = Identity on the quantity)   ( = Identity on the quantity) met, is bijective and is the inverse of, that is.
• The set of permutations of a given basic set, together with the composition as a link, a group, called the symmetric group of.

History

After you've been for generations got by with phrases such as " one to one ", came only in the mid-20th century by the solid- set-theoretic representation of all mathematical subdivisions on the need for a more concise description. Probably the word was coined injective as well as surjective and bijective in the 1930s by N. Bourbaki.