bijectivité

Bijectivité is a fundamental concept in mathematics, specifically within set theory and abstract algebra. A function f that maps elements from a set A to a set B is considered bijective if it possesses two key properties: injectivity and surjectivity.

Injectivity, also known as one-to-one, means that for every element in the codomain B, there is at most one element in the domain A that maps to it. In other words, distinct elements in A are mapped to distinct elements in B. If f(a1) = f(a2), then it must be that a1 = a2 for any a1, a2 in A.

Surjectivity, also known as onto, means that for every element in the codomain B, there is at least one element in the domain A that maps to it. Essentially, the range of the function f is equal to its codomain B. Every element in B is "hit" by at least one element from A.

When a function is both injective and surjective, it is called bijective. This means that there is a perfect pairing between the elements of the domain and the codomain. Each element in A maps to a unique element in B, and every element in B is mapped to by a unique element in A. This property implies that the two sets A and B have the same cardinality, meaning they have the same "size" in terms of the number of elements they contain. Bijective functions are crucial for establishing one-to-one correspondences and are essential in defining concepts like isomorphisms in various algebraic structures.