Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set is the image of exactly one element of the first set. 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 it is invertible; that is, a function is bijective if and only if there is a function the inverse of, such that each of the two ways for composing the two functions produces an identity function: for each in and for each in
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 —meaning that each element in the codomain is mapped from at most one element of the domain—and surjective —meaning that each element of the codomain is mapped from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers, 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, 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, permutations, 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 and means that a pairing 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

Every map from the empty set to itself is a bijection.

Batting line-up of a baseball or cricket team

Consider the batting line-up of a baseball or cricket team. The set X will be the players on the team and the set Y will be the 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 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:

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

For any set X, the identity function 1_X: X → X, 1_X = x is bijective.
The multiplicative inverse function gives a bijection of the unit interval with the semi-infinite interval.
The function f: R → R, f = 2x + 1 is bijective, since for each y there is a unique x = /2 such that f = y. More generally, any linear function over the reals, f: R → R, f = ax + b is a bijection. Each real number y is obtained from the real number x = /a.
The function f: R →, given by f = arctan is bijective, since each real number x is paired with exactly one angle y in the interval so that tan = 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 = e^x, is not bijective: for instance, there is no x in R such that g = −1, showing that g is not onto. However, if the codomain is restricted to the positive real numbers, then g would be bijective; its inverse is the natural logarithm function ln.
The function h: R → R⁺, h = x² is not bijective: for instance, h = h = 1, showing that h is not one-to-one. However, if the domain is restricted to, 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 also defines a converse 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
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 and 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 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, S_X, 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| and |f⁻¹| = |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!.
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.