In
mathematics, a bijection, or
a bijective function is a
function
f from a
set X
to a set Y with the property that, for every y in Y, there is
exactly one x in X such that f(x) = y.
Alternatively, f is bijective if it is a
one-to-one correspondence between those sets;
i.e., both
one-to-one (
injective)
and onto (
surjective).
(One-to-one function means one-to-one correspondence (i.e.,
bijection) to some authors, but injection to others.)
For example, consider the function succ, defined
from the set of
integers
\Z to \Z, that to each integer x associates the integer succ(x) = x
+ 1. For another example, consider the function sumdif that to each
pair (x,y) of real numbers associates the pair sumdif(x,y) =
(x + y, x − y).
A bijective function from a set to itself is also
called a
permutation.
The set of all bijections from X to Y is denoted
as X\leftrightarrowY.
Bijective functions play a fundamental role in
many areas of mathematics, for instance in the definition of
isomorphism (and
related concepts such as
homeomorphism and
diffeomorphism),
permutation
group,
projective
map, and many others.
Composition and inverses
A function f is bijective
if and
only if its
inverse
relation f −1 is a function. In that case, f
−1 is also a bijection.
The
composition
g o f of two bijections f\;:\; X\leftrightarrowY
and g\;:\; Y\leftrightarrowZ is a bijection. The inverse of
g o f is (g o f)−1 =
(f −1) o (g−1).
On the other hand, if the composition
g o f of two functions is bijective, we can only
say that f is injective and g is
surjective.
A relation f from X to Y is a bijective function
if and only if there exists another relation g from Y to X such
that g o f is the
identity
function on X, and f o g is the
identity
function on Y. Consequently, the sets have the same
cardinality.
Bijections and cardinality
If X and Y are
finite sets,
then there exists a bijection between the two sets X and Y
if and
only if X and Y have the same number of elements. Indeed, in
axiomatic
set theory, this is taken as the very definition of "same
number of elements", and generalising this definition to
infinite sets leads to the
concept of
cardinal
number, a way to distinguish the various sizes of
infinite
sets.
Bijections and category theory
Formally, bijections are
precisely the
isomorphisms in the
category
Set of
sets and functions. However,
the bijections are not always the isomorphisms. For example, in the
category
Top of
topological
spaces and
continuous
functions, the isomorphisms must be
homeomorphisms in addition
to being bijections.
bijection in Arabic: تقابل
bijection in Bulgarian: Биекция
bijection in Catalan: Funció bijectiva
bijection in Czech: Bijekce
bijection in Danish: Bijektiv
bijection in German: Bijektivität
bijection in Spanish: Función biyectiva
bijection in Esperanto: Ensurĵeto
bijection in French: Bijection
bijection in Korean: 전단사 함수
bijection in Croatian: Bijekcija
bijection in Ido: Bijektio
bijection in Icelandic: Gagntæk vörpun
bijection in Italian: Corrispondenza
biunivoca
bijection in Hebrew: פונקציה חד-חד-ערכית
ועל
bijection in Lithuanian: Bijekcija
bijection in Lombard: Bigezziú
bijection in Hungarian: Bijekció
bijection in Dutch: Bijectie
bijection in Japanese: 全単射
bijection in Norwegian: Bijeksjon
bijection in Norwegian Nynorsk: Bijeksjon
bijection in Occitan (post 1500):
Bijeccion
bijection in Polish: Funkcja wzajemnie
jednoznaczna
bijection in Portuguese: Função bijectiva
bijection in Russian: Биекция
bijection in Slovak: Bijektívne zobrazenie
bijection in Slovenian: Bijektivna
preslikava
bijection in Serbian: Бијекција
bijection in Finnish: Bijektio
bijection in Swedish: Bijektiv
bijection in Ukrainian: Бієкція
bijection in Chinese: 双射