thumb|200px|a bijective function, f: x â y, where set x is {1, 2, 3, 4} and set y is {a, b, c, d}. for example, f(1) = d. in mathematics, a bijection (or bijective function or one-to-one correspondence) is a function between the elements of two sets, where 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 â 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 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. 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 bijection 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).there are names associated to properties (1) and (2) as well. a relation which satisfies property (1) is called a total relation and a relation satisfying (2) is a single valued relation. 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 team=== consider the [[batting order 
Basic_concepts_in_set_theory Functions_and_mappings Mathematical_relations Types_of_functions