A bijection (or bijective function or one-to-one correspondence) is a function giving an exact pairing of the elements of two sets. Every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In formal mathematical terms, a bijective function f: X → Y is a one to one and onto mapping of a set X to a set Y.
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 complex, 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.
Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.
Read more about Bijection: Definition, Examples, More Mathematical Examples and Some Non-examples, Inverses, Composition, Bijections and Cardinality, Properties, Bijections and Category Theory