equipotenti

Equipotenti (equipotent) is a term used in set theory to describe sets that have the same cardinality. Two sets A and B are equipotent if there exists a bijection between them; in symbols, A and B are equipotent when there is a one-to-one correspondence f: A → B. This relation is an equivalence relation on the class of sets and partitions sets into equipotence classes, each corresponding to a cardinal number.

In finite sets, equipotence means simply that the sets contain the same number of elements. For example, {1, 2, 3} and {a, b, c} are equipotent, while {1, 2} is not equipotent to {a, b, c}.

Infinite sets reveal the central idea of cardinality. For infinite sets, equipotence means they have the same cardinality, even if one set is a proper subset of the other. Examples include the natural numbers N and the integers Z, which are equipotent; N and N×N are also equipotent; as are the open interval (0, 1) and the real numbers R. Distinctions between different infinite sizes are established by cardinal arithmetic and Cantor’s results; for instance, |N| < |R|, showing not all infinite sets are equipotent.

Key results include the Cantor–Bernstein–Schroeder theorem: if there are injections from A to B and from B to A, then A and B are equipotent. This underpins much of cardinal arithmetic and the formal treatment of cardinal numbers.

See also: cardinality, bijection, Cantor–Bernstein–Schroeder theorem, Cantor’s diagonal argument, cardinal numbers.