|x ↦ f (x)
|History of the function concept
|Examples of domains and codomains
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 is bijective if and only if there is a function the inverse of f, 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 (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, 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.
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold:
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). 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".
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.
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:
The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
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
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.
The composition of two bijections f: X → Y and g: Y → Z 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.
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.
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.
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.
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.
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.
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: