Equinumerosity

In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, 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.^[1] Equinumerous sets are said to have the same cardinality (number of elements).^[2] 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.

Equinumerosity has the characteristic properties of an equivalence relation.^[1] The statement that two sets A and B are equinumerous is usually denoted

${\displaystyle A\approx B\,}$ or ${\displaystyle A\sim B}$, or ${\displaystyle |A|=|B|.}$

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. 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 his 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 (an example where a proper subset of an infinite set is equinumerous to the original set), 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 (the set of all its subsets).^[1] This allows the definition of greater and greater infinite sets starting from a single infinite set.

If the axiom of choice holds, then the cardinal number of a set may be regarded as the least ordinal number of that cardinality (see initial ordinal). Otherwise, it may be regarded (by Scott's trick) as the set of sets of minimal rank having that cardinality.^[1]

The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the axiom of choice.^[3]