Bijection

In mathematics, a bijection, 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 f: XY is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.[1] The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures).

A bijective function, f: XY, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.

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.


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