Bijective function

A bijective function

Bijectivity (the adjective bijective comprising about, clearly reversibly on 'is - hence the term one-one or substantively corresponding to one correspondence ) is a mathematical term in the field of set theory . It describes a special property of images and functions . Bijective images and functions are also called bijections . Bijections associated with a mathematical structure often have their own names such as isomorphism , diffeomorphism , homeomorphism , mirroring or the like. As a rule, additional requirements have to be met here with regard to the maintenance of the structure under consideration.

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

In the case of a bijection , the definition set and the target set always have the same thickness . In the case that there is a bijection between two finite sets , this common cardinality is a natural number , namely exactly the number of elements in each of the two sets .

The bijection of a set onto itself is also called permutation . Here, too, there are many names of their own in mathematical structures. If the bijection has structure-preserving properties beyond that, it is called an automorphism .

A bijection between two sets is sometimes called a bijective correspondence .

definition

Be and sets and be a function that maps from to , that is . Then called bijective if for just one with exists. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f}$ ${\displaystyle y\in Y}$ ${\displaystyle x\in X}$ ${\displaystyle f\left(x\right)=y}$

That means: is bijective if and only if both ${\displaystyle f}$ ${\displaystyle f}$

(1) Injective is:

No value of the target quantity is assumed more than once . In other words: the archetype of each element of the target set consists of at most one element of . It therefore always follows from . ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle f(x_{1})=f(x_{2})}$ ${\displaystyle x_{1}=x_{2}}$

as well as

(2) is surjective :

Every element of the target set is accepted. In other words: the target set and the image set match, that is . For each out there is (at least) one out with . ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle f(X)}$ ${\displaystyle f\left(X\right)=Y}$ ${\displaystyle y}$ ${\displaystyle Y}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f(x)=y}$

Graphic illustrations

• 

  The principle of bijectivity: each point in the target set (Y) is hit exactly once.

• 

  Four bijective, strictly monotonically increasing, real continuous functions.

• 

  Four bijective strictly monotonically decreasing real continuous functions.

Examples and counterexamples

The set of real numbers is denoted by, the set of non-negative real numbers by . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} _{0}^{+}}$

• The function is bijective with the inverse function . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ,x\mapsto x+a}$ ${\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} ,x\mapsto x-a}$
• Likewise for the function is bijective with the inverse function . ${\displaystyle a\neq 0}$ ${\displaystyle g\colon \mathbb {R} \to \mathbb {R} ,x\mapsto ax}$ ${\displaystyle g^{-1}\colon \mathbb {R} \to \mathbb {R} ,x\mapsto {\frac {x}{a}}}$
• Example: If one assigns each ( monogamously ) married person to his or her spouse, this is a bijection of the set of all married people onto himself. This is even an example of a self-inverse mapping .

• The following four square functions only differ in their definition or value sets:

${\displaystyle f_{1}\colon \mathbb {R} \ \ \rightarrow \mathbb {R} ,\ \ \ x\mapsto x^{2}}$

${\displaystyle f_{2}\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} ,\ \ \ x\mapsto x^{2}}$

${\displaystyle f_{3}\colon \mathbb {R} \ \ \rightarrow \mathbb {R} _{0}^{+},\ x\mapsto x^{2}}$

${\displaystyle f_{4}\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} _{0}^{+},\ x\mapsto x^{2}}$

With these definitions is

${\displaystyle f_{1}}$ not injective, not surjective, not bijective

${\displaystyle f_{2}}$ injective, not surjective, not bijective

${\displaystyle f_{3}}$ not injective, surjective, not bijective

${\displaystyle f_{4}}$ injective, surjective, bijective

properties

• If and are finite sets with the same number of elements and is a function, then: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\colon A\to B}$
  • Is injective, then it is already bijective. ${\displaystyle f}$ ${\displaystyle f}$
  • If surjective is already bijective. ${\displaystyle f}$ ${\displaystyle f}$

• In particular, the following holds for functions of a finite set in themselves: ${\displaystyle f\colon A\to A}$ ${\displaystyle A}$
  • ${\displaystyle f}$is injective ⇔ is surjective ⇔ is bijective. ${\displaystyle f}$ ${\displaystyle f}$
  • This is generally wrong for infinite sets. These can be mapped injectively onto real subsets; there are also surjective mappings of an infinite set onto themselves that are not bijections. Such surprises are described in more detail in the Hilbert's Hotel article, see also Dedekind infinity .

• If the functions and are bijective, then this also applies to the concatenation . The inverse function of is then . ${\displaystyle f\colon A\to B}$ ${\displaystyle g\colon B\to C}$ ${\displaystyle g\circ f\colon A\to C}$ ${\displaystyle g\circ f}$ ${\displaystyle f^{-1}\circ g^{-1}}$

• Is bijective, then is injective and surjective. ${\displaystyle g\circ f}$ ${\displaystyle f}$ ${\displaystyle g}$

• Is a function and is there a function that the two equations ${\displaystyle f\colon A\to B}$ ${\displaystyle g\colon B\to A}$

${\displaystyle g\circ f=\operatorname {id} _{A}}$( = Identity on the crowd ) ${\displaystyle \operatorname {id} _{A}}$ ${\displaystyle A}$

${\displaystyle f\circ g=\operatorname {id} _{B}}$( = Identity on the crowd ) ${\displaystyle \operatorname {id} _{B}}$ ${\displaystyle B}$

is fulfilled, then is bijective, and is the inverse function of , thus . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g=f^{-1}}$

• The set of permutations of a given basic set , together with the composition as a link, forms a group , the so-called symmetrical group of . ${\displaystyle A}$ ${\displaystyle A}$

History of the term

After using formulations such as “one-to-one” for a long time, the need for a more concise description finally arose in the middle of the 20th century in the course of the consistent set-theoretical representation of all mathematical sub-areas. The terms bijective , injective and surjective were coined in the 1950s by the Nicolas Bourbaki group of authors .

literature

• Heinz-Dieter Ebbinghaus: Introduction to set theory . 4th edition. Spectrum Academic Publishing House, Heidelberg [u. a.] 2003, ISBN 3-8274-1411-3 .
• Gerd Fischer: Linear Algebra . 17th edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 .
• Walter Gellert, Herbert Kästner , Siegfried Neuber (Hrsg.): Fachlexikon ABC Mathematik . Verlag Harri Deutsch, Thun and Frankfurt / Main 1978, ISBN 3-87144-336-0 .

Web links

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

Individual evidence

1. ↑ Don Zagier : Zeta functions and quadratic fields: An introduction to higher number theory . Springer, 1981, ISBN 3-540-10603-0 , here p. 94 ( limited preview in Google book search [accessed June 7, 2017]).
2. ↑ Gernot Stroth : Algebra: Introduction to Galois Theory . de Gruyter, Berlin 1998, ISBN 3-11-015534-6 , here p. 100 ( limited preview in Google book search [accessed June 7, 2017]).
3. ^ Earliest Known Uses of Some of the Words of Mathematics.