Bijetiva

Bijetiva is a concept in mathematics that describes a one-to-one correspondence between two sets. A function f from a set A to a set B is bijective if it is both injective (one-to-one) and surjective (onto). Injective means that different elements of A map to different elements of B; surjective means that every element of B is the image of some element of A. When these conditions hold, there exists a unique inverse function f^{-1}: B -> A, satisfying f^{-1}(f(a)) = a for all a in A and f(f^{-1}(b)) = b for all b in B. A bijection thus pairs every element of A with exactly one element of B and vice versa, creating a one-to-one correspondence.

Bijective functions preserve cardinality: A and B have the same cardinality, often denoted |A| = |B|. If A and B are finite with the same size, a function between them is bijective if and only if it is injective or surjective.

Examples: The identity function id_A: A -> A is a bijection. The function f: {1,2,3} -> {a,b,c} with f(1)=a, f(2)=b, f(3)=c is a bijection. A typical non-bijective example is f: N -> N, f(n)=n^2, which is injective but not surjective; or f: R -> R, f(x)=x^2, which is neither injective nor surjective onto R.

Applications: bijections are central in counting arguments, in defining inverse functions, and in establishing equivalences between sets, as well as in many areas of algebra and analysis.