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In
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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
; see figures).
A bijection from the set ''X'' to the set ''Y'' has an
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
from ''Y'' to ''X''. If ''X'' and ''Y'' are
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. ...
s, then the existence of a bijection means they have the same number of elements. For
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only ...
s, 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. T ...
—a way to distinguish the various sizes of infinite sets.
A bijective function from a set to itself is also called a ''
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
'', and the set of all permutations of a set forms the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
.
Bijective functions are essential to many areas of mathematics including the definitions of
isomorphism,
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 isomor ...
,
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
,
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 ''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 function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
s" 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, unc ...
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
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
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
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
(−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
, then ''g'' would be bijective; its inverse (see below) is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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 is restricted to
, 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
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the fun ...
) 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
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
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 of two bijections ''f'': ''X → Y'' and ''g'': ''Y → Z'' is a bijection, whose inverse is given by
is
.
Conversely, if the composition
of two functions is bijective, it only follows that ''f'' is
injective and ''g'' is
surjective.
Cardinality
If ''X'' and ''Y'' are
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. ...
s, 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 bi ...
''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 set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only ...
s 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. T ...
, a way to distinguish the various sizes of infinite sets.
Properties
* A function ''f'': R → R is bijective if and only if its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
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
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
of ''X'', which is denoted variously by S(''X''), ''S
X'', or ''X''! (''X''
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
).
* 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 set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. ...
s 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
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
s of elements of ''S'' is the same as the number of total orderings of that set—namely, ''n''!.
Category theory
Bijections are precisely the
isomorphisms in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
''
Set'' 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, the morphisms must be
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
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, 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 __NOTOC__
In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for ...
.
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 o ...
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 injec ...
*
Bijection, injection and surjection
*
Bijective numeration
*
Bijective proof
*
Category theory
*
Multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
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:
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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