Bijektive

Bijektive, or bijective in English, describes a type of function in mathematics that is both injective (one-to-one) and surjective (onto). A function f: A → B is injective if distinct elements of A map to distinct elements of B, and it is surjective if every element of B is the image of some element of A. A bijection therefore creates a perfect pairing between the domain A and the codomain B.

Because a bijection pairs each element of A with exactly one element of B and covers all of B, it is invertible. There exists a unique 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 is itself a function, and the composition of bijections is a bijection.

In terms of cardinality, bijections preserve size. If A and B are finite, a bijection between them exists if and only if |A| = |B|. More generally, a bijection demonstrates that A and B have the same cardinality. When A = B, a bijection is a permutation of the set.

Typical examples include the identity function on any set, which is trivially bijective, and a simple one-to-one correspondence like f: {1,2,3} → {a,b,c} with f(1)=a, f(2)=b, f(3)=c. Non-bijective examples include maps that repeat outputs (not injective) or miss outputs in the codomain (not surjective). See also inverse function, injective, and surjective.