Bijection

Last updated April 27, 2024

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is mapped to from exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijective if and only if it is invertible; that is, a function $f:X\to Y$ is bijective if and only if there is a function $g:Y\to X,$ the inverse of $f$ , such that each of the two ways for composing the two functions produces an identity function: $g(f(x))=x$ for each $x$ in $X$ and $f(g(y))=y$ for each $y$ in $Y.$

For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.

A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped to from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped to from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function.

The elementary operation of counting establishes a bijection from some finite set to the first natural numbers $(1, 2, 3, ...)$ , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.

A bijective function from a set to itself is also called a permutation,^[1] and the set of all permutations of a set forms its symmetric group.

Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.

Definition

For a binary relation pairing elements of set X with elements of set Y 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 functions" and are called injections (or injective functions).^[2] 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".^[3]

Examples

Batting line-up of a baseball or cricket team

Consider the batting line-up of a baseball or cricket 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 1_X: X → X, 1_X(x) = x is bijective.
The function f: R → R, f(x) = 2x + 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, 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 $\mathbb {R} ^{+}\equiv \left(0,\infty \right)$ , then g would be bijective; its inverse (see below) is the natural logarithm function ln.
The function h: R → R⁺, h(x) = x² 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 $\mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)$ , then h would be bijective; its inverse is the positive square root function.
By Schröder–Bernstein 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 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 $(g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})$ .

Conversely, if the composition $g\,\circ \,f$ of two functions is bijective, it only follows that f is injective and g is surjective.

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 definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, 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), S_X, 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:
1. f is a bijection.
2. f is a surjection.
3. 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 isomorphisms in the category 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 homomorphisms 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.^[4]

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 of A and B′ is a subset of B.^[5]

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation .^[6] An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.^[7]

Gallery

An injective non-surjective function (injection, not a bijection)
An injective surjective function (bijection)
A non-injective surjective function (surjection, not a bijection)
A non-injective non-surjective function (also not a bijection)

Notes

↑ Hall 1959 , p. 3
↑ There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation and a relation satisfying (2) is a single valued relation.
↑ "Bijection, Injection, And Surjection | Brilliant Math & Science Wiki". brilliant.org. Retrieved 7 December 2019.
↑ Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
↑ Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7.
↑ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.
↑ John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.). Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4. preprint citing Lawson, M. V. (1998). "The Möbius Inverse Monoid". Journal of Algebra. 200 (2): 428–438. doi: 10.1006/jabr.1997.7242 .

Related Research Articles

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets $X$ and $Y$ is a set of ordered pairs $(x, y)$ consisting of elements $x$ from $X$ and $y$ from $Y$ . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element $x$ is related to an element $y$ , if and only if the pair $(x, y)$ belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case $n = 2$ of an $n$ -ary relation over sets $X 1, ..., X n$ , which is a subset of the Cartesian product

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number $is equal to itself (reflexive). If, then (symmetric). If and, then (transitive).$

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, the inverse function of a function $f$ is a function that undoes the operation of $f$ . The inverse of $f$ exists if and only if $f$ is bijective, and if it exists, is denoted by

In mathematics, a partial function $f$ from a set $X$ to a set $Y$ is a function from a subset $S$ of $X$ to $Y$ . The subset $S$ , that is, the domain of $f$ viewed as a function, is called the domain of definition or natural domain of $f$ . If $S$ equals $X$ , that is, if $f$ is defined on every element in $X$ , then $f$ is said to be a total function.

In mathematics, a surjective function (also known as surjection, or onto function ) is a function $f$ such that, for every element $y$ of the function's codomain, there exists at least one element $x$ in the function's domain such that $f (x) = y$ . In other words, for a function $f : X \to Y$ , the codomain $Y$ is the image of the function's domain $X$ . It is not required that $x$ be unique; the function $f$ may map one or more elements of $X$ to the same element of $Y$ .

In mathematics, an injective function (also known as injection, or one-to-one function ) is a function $f$ that maps distinct elements of its domain to distinct elements; that is, $x 1 \neq x 2$ implies $f (x 1) \neq f (x 2)$ . (Equivalently, $f (x 1) = f (x 2)$ implies $x 1 = x 2$ in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.

<span class="mw-page-title-main">Domain of a function</span> Mathematical concept

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by $or, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".$

<span class="mw-page-title-main">Codomain</span> Target set of a mathematical function

In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set $Y$ in the notation $f : X \to Y$ . The term range is sometimes ambiguously used to refer to either the codomain or the image of a function.

In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g₁, g₂: Y → Z,

In mathematics, a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$ . The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.

<span class="mw-page-title-main">Range of a function</span> Subset of a functions codomain

In mathematics, the range of a function may refer to either of two closely related concepts:

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function $is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.$

In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.

<span class="mw-page-title-main">Bijection, injection and surjection</span> Properties of mathematical functions

In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.

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:

Hall, Marshall Jr. (1959). The Theory of Groups. MacMillan.
Wolf (1998). Proof, Logic and Conjecture: A Mathematician's Toolbox. Freeman.
Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall.
Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole).
Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.
O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall.
Morash. Bridge to Abstract Mathematics. Random House.
Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.
Lay (2001). Analysis with an introduction to proof. Prentice Hall.
Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall.
Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent.
Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.
Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/ CRC Press.
D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.
Cupillari (1989). The Nuts and Bolts of Proofs . Wadsworth. ISBN 9780534103200.
Bond. Introduction to Abstract Mathematics. Brooks/Cole.
Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.
Ash. A Primer of Abstract Mathematics. MAA.

External links

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[1] Hall 1959 , p. 3

[2] There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation and a relation satisfying (2) is a single valued relation.

[3] "Bijection, Injection, And Surjection | Brilliant Math & Science Wiki". brilliant.org. Retrieved 7 December 2019.

[Hollings2014-4] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[Borceux1994-5] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7.

[Grillet1995-6] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.

[Campbell2007-7] John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.). Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4. preprint citing Lawson, M. V. (1998). "The Möbius Inverse Monoid". Journal of Algebra. 200 (2): 428–438. doi: 10.1006/jabr.1997.7242 .

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