bijekcj

Bijekcja, or bijection, is a concept in mathematics describing a special kind of function that pairs every element of one set with a unique element of another set in a way that covers the entire target set. Formally, a bijection f: A → B is both injective (one-to-one) and surjective (onto). Injective means different elements of A map to different elements of B; surjective means every element of B is the image of some element of A. Together, these conditions ensure a perfect pairing between A and B.

Key properties follow from this definition. A bijection has 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 is unique, and the existence of a bijection implies that A and B have the same cardinality. The inverse function also shows that bijections preserve structure in a strong sense and enable translating problems back and forth between the two sets.

Composition of bijections is a bijection: if f: A → B and g: B → C are bijections, then g ∘ f: A → C is a bijection. Examples include f(n) = n + 1 as a bijection from the integers to the integers, with inverse f⁻¹(n) = n − 1. A constant function, by contrast, is generally not a bijection unless its codomain contains a single element.

Bijekcje are central in set theory for comparing sizes of sets, including infinite ones, and in various areas of mathematics to show equivalences between structures.