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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
—a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a '' permutation'', and the set of all permutations of a set forms the symmetric group. Bijective functions are essential to many areas of mathematics including the definitions of
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
,
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
, diffeomorphism, permutation group, and
projective map In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ...
.


Definition

For a pairing between ''X'' and ''Y'' (where ''Y'' need not be different from ''X'') to be a bijection, four properties must hold: # each element of ''X'' must be paired with at least one element of ''Y'', # no element of ''X'' may be paired with more than one element of ''Y'', # each element of ''Y'' must be paired with at least one element of ''X'', and # no element of ''Y'' may be paired with more than one element of ''X''. Satisfying properties (1) and (2) means that a pairing is a function with
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
''X''. It is more common to see properties (1) and (2) written as a single statement: Every element of ''X'' is paired with exactly one element of ''Y''. Functions which satisfy property (3) are said to be " onto ''Y'' " and are called surjections (or ''surjective functions''). Functions which satisfy property (4) are said to be " one-to-one functions" and are called injections (or ''injective functions''). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto". Bijections are sometimes denoted by a two-headed rightwards arrow with tail (), as in ''f'' : ''X'' ⤖ ''Y''. This symbol is a combination of the two-headed rightwards arrow (), sometimes used to denote surjections, and the rightwards arrow with a barbed tail (), sometimes used to denote injections.


Examples


Batting line-up of a baseball or cricket team

Consider the batting line-up of a baseball or
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by st ...
team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set ''X'' will be the players on the team (of size nine in the case of baseball) and the set ''Y'' will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.


Seats and students of a classroom

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that: # Every student was in a seat (there was no one standing), # No student was in more than one seat, # Every seat had someone sitting there (there were no empty seats), and # No seat had more than one student in it. The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.


More mathematical examples

* For any set ''X'', the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
1''X'': ''X'' → ''X'', 1''X''(''x'') = ''x'' is bijective. * The function ''f'': R → R, ''f''(''x'') = 2''x'' + 1 is bijective, since for each ''y'' there is a unique ''x'' = (''y'' − 1)/2 such that ''f''(''x'') = ''y''. More generally, any linear function over the reals, ''f'': R → R, ''f''(''x'') = ''ax'' + ''b'' (where ''a'' is non-zero) is a bijection. Each real number ''y'' is obtained from (or paired with) the real number ''x'' = (''y'' − ''b'')/''a''. * The function ''f'': R → (−π/2, π/2), given by ''f''(''x'') = arctan(''x'') is bijective, since each real number ''x'' is paired with exactly one angle ''y'' in the interval (−π/2, π/2) so that tan(''y'') = ''x'' (that is, ''y'' = arctan(''x'')). If the codomain (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function. * The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, ''g'': R → R, ''g''(''x'') = e''x'', is not bijective: for instance, there is no ''x'' in R such that ''g''(''x'') = −1, showing that ''g'' is not onto (surjective). However, if the codomain is restricted to the positive real numbers \R^+ \equiv \left(0, \infty\right), then ''g'' would be bijective; its inverse (see below) is the natural logarithm function ln. * The function ''h'': R → R+, ''h''(''x'') = ''x''2 is not bijective: for instance, ''h''(−1) = ''h''(1) = 1, showing that ''h'' is not one-to-one (injective). However, if the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
is restricted to \R^+_0 \equiv \left ,_\infty\right),_then_''h''_would_be_bijective;_its_inverse_is_the_positive_square_root_function. *By_Cantor-Bernstein-Schröder_theorem,_given_any_two_sets_''X''_and_''Y'',_and_two_injective_functions_''f'':_''X_→_Y''_and_''g'':_''Y_→_X'',_there_exists_a_bijective_function_''h'':_''X_→_Y''.


_Inverses

A_bijection_''f''_with_domain_''X''_(indicated_by_''f'':_''X_→_Y''_in_Function_(mathematics)#Notation.html" ;"title="Cantor-Bernstein-Schröder_theorem.html" ;"title=", \infty\right), then ''h'' would be bijective; its inverse is the positive square root function. *By Cantor-Bernstein-Schröder theorem">, \infty\right), then ''h'' would be bijective; its inverse is the positive square root function. *By Cantor-Bernstein-Schröder theorem, given any two sets ''X'' and ''Y'', and two injective functions ''f'': ''X → Y'' and ''g'': ''Y → X'', there exists a bijective function ''h'': ''X → Y''.


Inverses

A bijection ''f'' with domain ''X'' (indicated by ''f'': ''X → Y'' in Function (mathematics)#Notation">functional notation) also defines a converse relation starting in ''Y'' and going to ''X'' (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, ''in general'', yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain ''Y''. Moreover, properties (1) and (2) then say that this inverse ''function'' is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection. Stated in concise mathematical notation, a function ''f'': ''X → Y'' is bijective if and only if it satisfies the condition :for every ''y'' in ''Y'' there is a unique ''x'' in ''X'' with ''y'' = ''f''(''x''). Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.


Composition

The composition g \,\circ\, f of two bijections ''f'': ''X → Y'' and ''g'': ''Y → Z'' is a bijection, whose inverse is given by g \,\circ\, f is (g \,\circ\, f)^ \;=\; (f^) \,\circ\, (g^). Conversely, if the composition g \, \circ\, f of two functions is bijective, it only follows that ''f'' is injective and ''g'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
.


Cardinality

If ''X'' and ''Y'' are finite sets, then there exists a bijection between the two sets ''X'' and ''Y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''X'' and ''Y'' have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" ( equinumerosity), and generalising this definition to infinite sets leads to the concept of
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
, a way to distinguish the various sizes of infinite sets.


Properties

* A function ''f'': R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once. * If ''X'' is a set, then the bijective functions from ''X'' to itself, together with the operation of functional composition (∘), form a group, the symmetric group of ''X'', which is denoted variously by S(''X''), ''SX'', or ''X''! (''X'' factorial). * Bijections preserve cardinalities of sets: for a subset ''A'' of the domain with cardinality , ''A'', and subset ''B'' of the codomain with cardinality , ''B'', , one has the following equalities: *:, ''f''(''A''), = , ''A'', and , ''f''−1(''B''), = , ''B'', . *If ''X'' and ''Y'' are finite sets with the same cardinality, and ''f'': ''X → Y'', then the following are equivalent: *# ''f'' is a bijection. *# ''f'' is a surjection. *# ''f'' is an injection. *For a finite set ''S'', there is a bijection between the set of possible total orderings of the elements and the set of bijections from ''S'' to ''S''. That is to say, the number of permutations of elements of ''S'' is the same as the number of total orderings of that set—namely, ''n''!.


Category theory

Bijections are precisely the
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s in the category ''
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
'' of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category '' Grp'' of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, the morphisms must be
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s since they must preserve the group structure, so the isomorphisms are ''group isomorphisms'' which are bijective homomorphisms.


Generalization to partial functions

The notion of one-to-one correspondence generalizes to
partial functions In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is d ...
, where they are called ''partial bijections'', although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup. Another way of defining the same notion is to say that a partial bijection from ''A'' to ''B'' is any relation ''R'' (which turns out to be a partial function) with the property that ''R'' is the graph of a bijection ''f'':''A′''→''B′'', where ''A′'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''A'' and ''B′'' is a subset of ''B''. When the partial bijection is on the same set, it is sometimes called a ''one-to-one partial transformation''. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.preprint
citing


Gallery


See also

*
Ax–Grothendieck theorem In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck. The theorem is often given as this special case: If ''P'' is an inj ...
* Bijection, injection and surjection * Bijective numeration * Bijective proof *
Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
* Multivalued function


Notes


References

This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these: * * * * * * * * * * * * * * * * *


External links

* *
Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
{{Mathematical logic Functions and mappings Basic concepts in set theory Mathematical relations Types of functions