bijective

A bijective function, also known simply as a bijection, is a type of mathematical function that establishes a perfect pairing between elements of two sets. Specifically, a function \(f: A \to B\) is called bijective if it is both injective (one-to-one) and surjective (onto).

Injectivity ensures that no two distinct elements in the domain \(A\) map to the same element in the codomain \(B\). In other words, if \(f(a_1) = f(a_2)\), then \(a_1 = a_2\). Surjectivity guarantees that every element in the codomain \(B\) has at least one pre-image in the domain \(A\), meaning for every \(b \in B\), there exists an \(a \in A\) such that \(f(a) = b\).

A bijective function thus creates a one-to-one correspondence between the sets \(A\) and \(B\). This property is fundamental in various areas of mathematics, such as set theory, algebra, and analysis, because it implies that the two sets are of equivalent size or cardinality. When a bijection exists, the sets can be considered structurally similar or interchangeable for certain purposes, like defining inverse functions.

The inverse of a bijective function exists; it reverses the mapping, with \(f^{-1}: B \to A\), such that \(f^{-1}(f(a)) = a\) for all \(a \in A\). This concept is crucial in many mathematical proofs and applications, including calculus, combinatorics, and computer science, particularly in permutations and data encryption.

In summary, a bijective function is a perfect pairing between sets, characterized by its injective and surjective properties, enabling one-to-one correspondence and the existence of an inverse function.