Biunivocality

Biunivocality is a term used in logic and mathematics to describe a specific type of relationship between two sets. It denotes a one-to-one correspondence where each element in the first set is uniquely associated with exactly one element in the second set, and conversely, each element in the second set is uniquely associated with exactly one element in the first set. This is often referred to as a bijection.

For a relationship to be biunivocal, two conditions must be met: it must be injective (one-to-one) and surjective (onto). An injective function ensures that no two distinct elements in the first set map to the same element in the second set. A surjective function ensures that every element in the second set is mapped to by at least one element from the first set. When both these properties hold, the mapping is biunivocal.

The concept of biunivocality is fundamental in understanding equivalences between sets, particularly in cardinality. If a biunivocal relationship exists between two sets, it means they have the same number of elements, or the same cardinality. This is crucial in set theory, as it allows mathematicians to compare the sizes of infinite sets. For example, the set of natural numbers and the set of even numbers are biunivocal, indicating they have the same infinite cardinality. The term itself emphasizes the "two-way" or "mutual" nature of the one-to-one correspondence.