bijektivna

Bijektivna is a term used in several Slavic languages to describe a property of a function, map, or transformation. In English, it corresponds to the word bijective. A bijektivna funkcija (bijective function) is one that is both injective (one-to-one) and surjective (onto). Equivalently, it establishes a one-to-one correspondence between the elements of its domain and codomain.

A function f: A -> B is bijektivna if for every y in B there exists a unique x in A with f(x) = y, and no two distinct elements of A share the same image. This implies that f has an inverse function f^{-1}: B -> A, which is also bijektivna. The inverse undoes the mapping: f^{-1}(y) = x whenever f(x) = y.

Examples include f(x) = x + 1 from the real numbers to the real numbers, which is bijektivna; the linear function f(x) = ax + b with a ≠ 0 on R is bijektivna; and any permutation of a finite set is bijektivna.

Consequences of bijectivity include the equality of cardinalities of domain and codomain and invertibility of the map. In abstract algebra, a bijektivna homomorphism is an isomorphism, meaning it preserves structure. Not every bijection is a homomorphism, but when it is, it preserves the relevant algebraic operations.

See also: injection, surjection, bijection, inverse function, permutation.