Editing Bijection (section)

{{short description|One-to-one correspondence}}
{{redirect|One-to-one correspondence|one-to-one function|injective function}}
{{Use dmy dates|date=July 2022}}
[[Image:Bijection.svg|thumb|A bijective function, ''f'': ''X'' → ''Y'', where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, ''f''(1) = D.]]
{{Functions}}

In [[mathematics]], a '''bijection''', '''bijective function''', or '''one-to-one correspondence''' is a [[function (mathematics)|function]] between two [[set (mathematics)|sets]] such that each element of the second set (the [[codomain]]) is the image of exactly one element of the first set (the [[domain of a function|domain]]). Equivalently, a bijection is a [[binary relation|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 [[inverse function|invertible]]; that is, a function <math>f:X\to Y</math> is bijective if and only if there is a function <math>g:Y\to X,</math> the ''inverse'' of {{mvar|f}}, such that each of the two ways for [[function composition|composing]] the two functions produces an [[identity function]]: <math>g(f(x)) = x</math> for each <math>x</math> in <math>X</math> and <math>f(g(y)) = y</math> for each <math>y</math> in <math>Y.</math>

For example, the ''multiplication by two'' defines a bijection from the [[integer]]s to the [[even number]]s, 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 number]]s {{math|(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]],<ref>{{harvnb|Hall|1959|p=3}}</ref> and the set of all permutations of a set forms its [[symmetric group]].

Some bijections with further properties have received specific names, which include [[automorphism]]s, [[isomorphism]]s, [[homeomorphism]]s, [[diffeomorphism]]s, [[permutation group]]s, and most [[geometric transformation]]s. [[Galois correspondence]]s are bijections between sets of [[mathematical object]]s of apparently very different nature.