In mathematics, a bijection, bijective function or one-to-one correspondence is
a function between the elements of two sets, where every element of one set is
paired with exactly one element of the other set, and every 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 f: X → Y is
a one-to-one and onto mapping of a set X to a set Y.
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, a way to distinguish
the various sizes of infinite sets. A bijective function from a set to
itself is also called a permutation. Bijective functions are essential to
many areas of mathematics including the definitions of isomorphism,
homeomorphism, diffeomorphism, permutation group, and projective map.
Definition For a pairing between X and 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 and means that a
bijection is a function with domain X. It is more common to see properties and
written as a single statement: Every element of X is paired with exactly one
element of Y. Functions which satisfy property are said to be "onto Y " and
are called surjections. Functions which satisfy property are said to be
"one-to-one functions" and are called injections. 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".
Examples = Batting line-up of a baseball team=
Consider the batting line-up of a baseball team. The set X will be the
nine players on the team and the set Y will be the nine positions in the
batting order The "pairing" is given by which player is in what position in this
order. Property is satisfied since each player is somewhere in the list.
Property is satisfied since no player bats in two positions in the order.
Property says that for each position in the order, there is some player batting
in that position and property 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 all 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, No student was in more than one seat,
Every seat had someone sitting there, 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 1X:
X → X, 1X(x) = x, is bijective. The function f: R → R, f(x) = 2x + 1 is
bijective, since for each y there is a unique x =/2 such that f(x) = y. In more
generality, any linear function over the reals, f: R → R, f(x) = ax + b is a
bijection. Each real number y is obtained from the real number x =/a.
The function f: R →, given by f(x) = arctan(x) is bijective since each real
number x is paired with exactly one angle y in the interval so that tan(y) =
x). If the codomain was made larger to include an integer multiple of π/2 then
this function would no longer be onto 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) = ex, is not bijective: for instance,
there is no x in R such that g(x) = −1, showing that g is not onto. However if
the codomain is restricted to the positive real numbers , then g becomes
bijective; its inverse 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. However, if the domain is restricted to
, then h becomes bijective; its inverse is the positive square root function.
Inverses A bijection f with domain X also defines
a relation starting in Y and going to X. The process of "turning the arrows
around" for an arbitrary function does not, in general, yield a function, but
properties and of a bijection say that this inverse relation is a function with
domain Y. Moreover, properties and 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 of two bijections f: X → Y and g: Y → Z is a bijection. The
inverse of is . Conversely, if the composition of two
functions is bijective, we can only say that f is injective and g is surjective.
Bijections and 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", 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), SX, or X!.
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!. Bijections and 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-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 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 with the property that R is the graph of a
bijection f:A′→B′, where A′ is a subset of A and likewise B′⊆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. Contrast with
Multivalued function See also
Injective function Surjective function
Bijection, injection and surjection Symmetric group
Bijective numeration Bijective proof
Cardinality Category theory
Ax–Grothendieck theorem 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:
Wolf. Proof, Logic and Conjecture: A Mathematician's Toolbox. Freeman.
Sundstrom. Mathematical Reasoning: Writing and Proof. Prentice-Hall.
Smith; Eggen; St.Andre. A Transition to Advanced Mathematics. Thomson.
Schumacher. Chapter Zero: Fundamental Notions of Abstract Mathematics.
Addison-Wesley. O'Leary. The Structure of Proof: With
Logic and Set Theory. Prentice-Hall. Morash. Bridge to Abstract Mathematics.
Random House. Maddox. Mathematical Thinking and
Writing. Harcourt/ Academic Press. Lay. Analysis with an introduction to
proof. Prentice Hall. Gilbert; Vanstone. 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. Sets, Functions, and Logic: An Introduction to Abstract
Mathematics. Chapman & Hall/ CRC Press. D'Angelo; West. Mathematical Thinking:
Problem Solving and Proofs. Prentice Hall.
Cupillari. The Nuts and Bolts of Proofs. Wadsworth.
Bond. Introduction to Abstract Mathematics. Brooks/Cole.
Barnier; Feldman. Introduction to Advanced Mathematics. Prentice Hall.
Ash. A Primer of Abstract Mathematics. MAA.
External links Hazewinkel, Michiel, ed., "Bijection",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Bijection", MathWorld.
Earliest Uses of Some of the Words of Mathematics: entry on Injection,
Surjection and Bijection has the history of Injection and related terms.