Bijectives

A bijective function, or a bijection, is a function that is both injective (one-to-one) and surjective (onto). If f maps from a domain A to a codomain B and is bijective, every element of A is paired with a unique element of B, and every element of B is the image of some element of A. This creates a one-to-one correspondence between A and B.

Because every element of B is hit by the function, a bijection implies that A and B have the same cardinality, meaning they can be put into a perfect pairing with no leftovers. The term bijection is often described as a one-to-one correspondence between the two sets.

A bijection has a well-defined inverse. If f: A -> B is bijective, there exists an inverse function f^{-1}: B -> A such that f^{-1}(f(a)) = a for all a in A and f(f^{-1}(b)) = b for all b in B. The inverse of a bijection is also bijective.

Examples illustrate the concept. The function f(x) = x^3 from the real numbers to the real numbers is bijective. The function f(n) = n+1 from the integers to the integers is bijective as well. In a finite setting, any permutation of a finite set is a bijection. A non-example is f(n) = 2n from the integers to the integers, which is injective but not surjective.

Key properties include that the composition of bijections is a bijection, and the inverse of a bijection is a bijection. Bijective functions are central in discussions of cardinality, isomorphisms in set theory, and many areas of combinatorics and analysis.