Equipollence

Equipollence is a mathematical relation between sets that expresses having the same cardinality. Two sets A and B are equipollent if there exists a bijection from A to B. Equivalently, A and B are equipollent if they are equinumerous, meaning they contain the same number of elements in the sense of set theory, even when the sets are infinite.

The term appears in historical literature on the foundations of set theory, though the more common modern terminology is equinumerous or “having the same cardinality.” Equipollence is thus largely synonymous with having identical cardinal numbers, and it underlies the formal treatment of cardinalities in Cantor’s theory.

Examples help illustrate the concept. The set of natural numbers N and the set of even numbers 2N are equipollent via the bijection f(n) = 2n. The set of integers Z is equipollent to N, with a standard bijection that interleaves nonnegative and negative integers (for instance, mapping 0 to 0, 1 to 1, 2 to −1, 3 to 2, 4 to −2, and so on).

Formally, equipollence is an equivalence relation on the class of sets: it is reflexive, symmetric, and transitive. It partitions sets into cardinality classes and provides a way to compare the sizes of both finite and infinite sets. See also equinumerous, bijection, and cardinality.