Equipotent

In mathematics, two sets A and B are equinumerous if there exists a one-to-one correspondence (a bijection) between them, i.e. if there exists a function from A to B such that for every element y of B there is exactly one element x of A with f(x) = y. Equinumerous finite sets have the same number of elements. The definition of equinumerosity using bijections can be applied to both finite and infinite sets and allows one to state whether two sets have the same size even if they are infinite. Unlike finite sets, some infinite sets are equinumerous to proper subsets of themselves.

Equinumerous sets are said to have the same cardinality. The study of cardinality is often called equinumerosity (equalness-of-number). The terms equipollence (equalness-of-strength) and equipotence (equalness-of-power) are sometimes used instead. The statement that two sets A and B are equinumerous is usually denoted

Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof). In a controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous, and that the Cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers. Cantor's theorem from 1891 implies that no set is equinumerous to its own power set. This allows the definition of greater and greater infinite sets starting from a single infinite set.

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