Bijective function

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 (the codomain) is the image of 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 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 , 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 from at most one element of the domain—and surjective (or ''onto'')—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, 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''). 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 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 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 (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'') = 2''x'' + 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 $\R^+ \equiv \left(0, \infty\right)$, then ''g'' would be bijective; its inverse (see below) is the natural logarithm function ln.
* The function ''h'': R → R⁺, ''h''(''x'') = ''x''² 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 \R^+_0 \equiv \left , \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 Function (mathematics)#Notation">functional notation