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Bijection

From Wikipedia, the free encyclopedia
One-to-one correspondence

A bijective function,f:XY, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example,f(1) = D.
Function
xf (x)
History of the function concept
Types bydomain andcodomain
Classes/properties
  Constructions
  Generalizations  
  List of specific functions

Inmathematics, abijection,bijective function, orone-to-one correspondence is afunction between twosets such that each element of the second set (thecodomain) is the image of exactly one element of the first set (thedomain). Equivalently, a bijection is arelation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijectiveif and only if it isinvertible; that is, a functionf:XY{\displaystyle f:X\to Y} is bijective if and only if there is a functiong:YX,{\displaystyle g:Y\to X,} theinverse off, such that each of the two ways forcomposing the two functions produces anidentity function:g(f(x))=x{\displaystyle g(f(x))=x} for eachx{\displaystyle x} inX{\displaystyle X} andf(g(y))=y{\displaystyle f(g(y))=y} for eachy{\displaystyle y} inY.{\displaystyle Y.}

For example, themultiplication by two defines a bijection from theintegers to theeven numbers, which has thedivision by two as its inverse function.

A function is bijective if and only if it is bothinjective (orone-to-one)—meaning that each element in the codomain is mapped from at most one element of the domain—andsurjective (oronto)—meaning that each element of the codomain is mapped from at least one element of the domain. The termone-to-one correspondence must not be confused withone-to-one function, which means injective but not necessarily surjective.

The elementary operation ofcounting establishes a bijection from somefinite set to the firstnatural 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 samecardinal number if there exists a bijection between them.

A bijective function from a set to itself is also called apermutation,[1] and the set of all permutations of a set forms itssymmetric group.

Some bijections with further properties have received specific names, which includeautomorphisms,isomorphisms,homeomorphisms,diffeomorphisms,permutation groups, and mostgeometric transformations.Galois correspondences are bijections between sets ofmathematical objects of apparently very different nature.

Definition

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For abinary relation pairing elements of setX with elements of setY to be a bijection, four properties must hold:

  1. each element ofX must be paired with at least one element ofY,
  2. no element ofX may be paired with more than one element ofY,
  3. each element ofY must be paired with at least one element ofX, and
  4. no element ofY may be paired with more than one element ofX.

Satisfying properties (1) and (2) means that a pairing is afunction withdomainX. It is more common to see properties (1) and (2) written as a single statement: Every element ofX is paired with exactly one element ofY. Functions which satisfy property (3) are said to be "ontoY " and are calledsurjections (orsurjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are calledinjections (orinjective 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

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Batting line-up of a baseball or cricket team

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Consider thebatting line-up of a baseball orcricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The setX will be the players on the team (of size nine in the case of baseball) and the setY 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

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In a classroom there are a certain number of seats. A group 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:

  1. Every student was in a seat (there was no one standing),
  2. No student was in more than one seat,
  3. Every seat had someone sitting there (there were no empty seats), and
  4. 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

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A bijection from thenatural numbers to theintegers, which maps 2n to −n and 2n − 1 ton, forn ≥ 0.
  • For any setX, theidentity function1X:XX,1X(x) =x is bijective.
  • The functionf:RR,f(x) = 2x + 1 is bijective, since for eachy there is a uniquex = (y − 1)/2 such thatf(x) =y. More generally, anylinear function over the reals,f:RR,f(x) =ax +b (wherea is non-zero) is a bijection. Each real numbery is obtained from (or paired with) the real numberx = (yb)/a.
  • The functionf:R → (−π/2, π/2), given byf(x) = arctan(x) is bijective, since each real numberx is paired with exactly one angley in the interval (−π/2, π/2) so that tan(y) =x (that is,y = arctan(x)). If thecodomain (−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.
  • Theexponential function,g:RR,g(x) = ex, is not bijective: for instance, there is nox inR such thatg(x) = −1, showing thatg is not onto (surjective). However, if the codomain is restricted to the positive real numbersR+(0,){\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}, theng would be bijective; its inverse (see below) is thenatural logarithm function ln.
  • The functionh:RR+,h(x) =x2 is not bijective: for instance,h(−1) =h(1) = 1, showing thath is not one-to-one (injective). However, if thedomain is restricted toR0+[0,){\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}, thenh would be bijective; its inverse is the positive square root function.
  • BySchröder–Bernstein theorem, given any two setsX andY, and two injective functionsf:X → Y andg:Y → X, there exists a bijective functionh:X → Y.

Inverses

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A bijectionf with domainX (indicated byf:X → Y infunctional notation) also defines aconverse relation starting inY and going toX (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 domainY. Moreover, properties (1) and (2) then say that this inversefunction is a surjection and an injection, that is, theinverse function exists and is also a bijection. Functions that have inverse functions are said to beinvertible. A function is invertible if and only if it is a bijection.

Stated in concise mathematical notation, a functionf:X → Y is bijective if and only if it satisfies the condition

for everyy inY there is a uniquex inX withy =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

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A bijection composed of an injection (X → Y) and a surjection (Y → Z).

Thecompositiongf{\displaystyle g\,\circ \,f} of two bijectionsf:X → Y andg:Y → Z is a bijection, whose inverse is given bygf{\displaystyle g\,\circ \,f} is(gf)1=(f1)(g1){\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}.

Conversely, if the compositiongf{\displaystyle g\,\circ \,f} of two functions is bijective, it only follows thatf isinjective andg issurjective.

Cardinality

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IfX andY arefinite sets, then there exists a bijection between the two setsX andYif and only ifX andY have the same number of elements. Indeed, inaxiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition toinfinite sets leads to the concept ofcardinal number, a way to distinguish the various sizes of infinite sets.

Properties

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  • A functionf:RR is bijective if and only if itsgraph meets every horizontal and vertical line exactly once.
  • IfX is a set, then the bijective functions fromX to itself, together with the operation of functional composition (∘), form agroup, thesymmetric group ofX, which is denoted variously by S(X),SX, orX! (Xfactorial).
  • Bijections preservecardinalities of sets: for a subsetA of the domain with cardinality |A| and subsetB of the codomain with cardinality |B|, one has the following equalities:
    |f(A)| = |A| and |f−1(B)| = |B|.
  • IfX andY arefinite sets with the same cardinality, andf:X → Y, then the following are equivalent:
    1. f is a bijection.
    2. f is asurjection.
    3. f is aninjection.
  • For a finite setS, there is a bijection between the set of possibletotal orderings of the elements and the set of bijections fromS toS. That is to say, the number ofpermutations of elements ofS is the same as the number of total orderings of that set—namely,n!.

Category theory

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Bijections are precisely theisomorphisms in thecategorySet ofsets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the categoryGrp ofgroups, the morphisms must behomomorphisms since they must preserve the group structure, so the isomorphisms aregroup isomorphisms which are bijective homomorphisms.

Generalization to partial functions

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The notion of one-to-one correspondence generalizes topartial functions, where they are calledpartial 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 atotal function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called thesymmetric inverse semigroup.[4]

Another way of defining the same notion is to say that a partial bijection fromA toB is any relationR (which turns out to be a partial function) with the property thatR is thegraph of a bijectionf:A′B′, whereA′ is asubset ofA andB′ is a subset ofB.[5]

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

Gallery

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

See also

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Notes

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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 atotal relation and a relation satisfying (2) is asingle valued relation.
  3. ^"Bijection, Injection, And Surjection | Brilliant Math & Science Wiki".brilliant.org. Retrieved7 December 2019.
  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.
  5. ^Francis Borceux (1994).Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289.ISBN 978-0-521-44179-7.
  6. ^Pierre A. Grillet (1995).Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228.ISBN 978-0-8247-9662-4.
  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 citingLawson, M. V. (1998)."The Möbius Inverse Monoid".Journal of Algebra.200 (2):428–438.doi:10.1006/jabr.1997.7242.

References

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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.

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