TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a bijection, bijective function, one-to-one correspondence, or invertible function, is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 and 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 , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...

; see figures). A bijection from the set ''X'' to the set ''Y'' has an
inverse function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
from ''Y'' to ''X''. If ''X'' and ''Y'' are
finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, then the existence of a bijection means they have the same number of elements. For
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s, the picture is more complicated, leading to the concept of
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
—a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a ''
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

'', and the set of all permutations of a set forms a
symmetry group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...
. Bijective functions are essential to many areas of mathematics including the definitions of
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

,
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
,
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and
projective map In projective geometry, a homography is an isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
.

# 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 *Doma ...
''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 In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

''Y'' " and are called (or ''surjective functions''). Functions which satisfy property (4) are said to be "
one-to-one function In , an injective function (also known as injection, or one-to-one function) is a that maps elements to distinct elements; that is, implies . In other words, every element of the function's is the of one element of its . The term must no ...
s" and are called (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 Baseball is a bat-and-ball games, bat-and-ball game played between two opposing teams who take turns batting (baseball), batting and fielding. The game proceeds when a player on the fielding team (baseball), fielding team, called the pi ...

or
cricket Cricket is a bat-and-ball gameBat-and-ball may refer to: *Bat-and-ball games Bat-and-ball games (or safe haven games) are field games played by two opposing teams, in which the action starts when the defending team throws a ball at a ded ...

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 and some non-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 (mathematics), function that always returns the same value that was ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(−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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of a ...

, ''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 $\scriptstyle \R^+ \;\equiv\; \left\left(0,\, +\infty\right\right)$, 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 (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...
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 *Doma ...
is restricted to ,_then_''h''_would_be_bijective;_its_inverse_is_the_positive_square_root_function. *By_Cantor-Bernstein-Schroder_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-Schroder_theorem.html" ;"title=",\, +\infty\right), then ''h'' would be bijective; its inverse is the positive square root function. *By Cantor-Bernstein-Schroder theorem">,\, +\infty\right), then ''h'' would be bijective; its inverse is the positive square root function. *By Cantor-Bernstein-Schroder 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
$\scriptstyle g \,\circ\, f$ of two bijections ''f'': ''X → Y'' and ''g'': ''Y → Z'' is a bijection, whose inverse is given by $\scriptstyle g \,\circ\, f$ is $\scriptstyle \left(g \,\circ\, f\right)^ \;=\; \left(f^\right) \,\circ\, \left(g^\right)$. Conversely, if the composition $\scriptstyle g \, \circ\, f$ of two functions is bijective, it only follows that ''f'' is
injective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and ''g'' is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.

# Cardinality

If ''X'' and ''Y'' are
finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, then there exists a bijection between the two sets ''X'' and ''Y''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
''X'' and ''Y'' have the same number of elements. Indeed, in
axiomatic set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although ob ...
, this is taken as the definition of "same number of elements" (
equinumerosity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), and generalising this definition to
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s leads to the concept of
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
, 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 discret ...

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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
of ''X'', which is denoted variously by S(''X''), ''SX'', or ''X''! (''X''
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
). * Bijections preserve
cardinalities In mathematics, the cardinality of a set (mathematics), set is a measure of the "number of Element (mathematics), elements" of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 1 ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s with the same cardinality, and ''f'': ''X → Y'', then the following are equivalent: *# ''f'' is a bijection. *# ''f'' is a
surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. *# ''f'' is an
injection Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...
. *For a finite set ''S'', there is a bijection between the set of possible
total ordering In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some Set (mathematics), set X, which is Antisymmetric relation, antisymmetric, Transitive relation, transitive, and a connex relation. ...
s of the elements and the set of bijections from ''S'' to ''S''. That is to say, the number of
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

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
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
'' 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 A group is a number of people 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 identi ...
, the morphisms must be
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

, 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 (mathematics), 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 ...
. 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 a bijection ''f'':''A′''→''B′'', where ''A′'' is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transf ...
''. An example is the
Möbius transformation In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
simply defined on the complex plane, rather than its completion to the extended complex plane.preprint
citing

# Contrast with

*
Multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

*
Bijection, injection and surjection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Bijective numeration Bijective numeration is any numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning " ...
*
Bijective proofIn combinatorics, bijective proof is a mathematical proof, proof technique that finds a bijective function (that is, a Injective function, one-to-one and onto function) between two finite sets and , or a size-preserving bijective function between t ...
*
Category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
* Ax–Grothendieck theorem

# 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: * * * * * * * * * * * * * * * * *