Bijektiflik

Bijektiflik, also known as a bijective function or one-to-one correspondence, is a fundamental concept in mathematics, particularly in set theory and abstract algebra. A function f from a set A to a set B is bijective if and only if it is both injective (one-to-one) and surjective (onto).

Injectivity means that distinct elements in the domain set A are mapped to distinct elements in the codomain set B. In other words, if f(x) = f(y), then x must equal y for all x and y in A. This ensures that no two different inputs produce the same output.

Surjectivity means that every element in the codomain set B has at least one element in the domain set A that maps to it. For every y in B, there exists an x in A such that f(x) = y. This implies that the range of the function is equal to its codomain.

When a function is bijective, it establishes a perfect pairing between the elements of the two sets. This means that the number of elements in set A is equal to the number of elements in set B. This property is crucial for defining concepts such as isomorphisms in various algebraic structures and for establishing equivalences between different mathematical objects. Bijective functions are also invertible, meaning that an inverse function exists that can reverse the mapping.