Bijective Map

In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each 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 is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures).

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'', and the set of all permutations of a set forms the symmetric group.

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'' (where ''Y'' need not be different from ''X'') 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".

Bijections are sometimes denoted by a two-headed rightwards arrow with tail (), as in ''f'' : ''X'' ⤖ ''Y''. This symbol is a combination of the two-headed rightwards arrow (), sometimes used to denote surjections, and the rightwards arrow with a barbed tail (), sometimes used to denote injections.

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 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:
# 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_Cantor-Bernstein-Schröder_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

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*By Cantor-Bernstein-Schröder theorem">, \infty\right), then ''h'' would be bijective; its inverse is the positive square root function.
*By Cantor-Bernstein-Schröder 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