Equipotent

Equipotent is a term used in set theory to describe two sets that have the same cardinality. Two sets A and B are equipotent if there exists a bijection f: A → B, meaning every element of A is paired with a unique element of B and vice versa. When this happens, the sets are said to be equinumerous or to have equal power, and one writes |A| = |B|.

For finite sets, equipotence is equivalent to having the same number of elements. For example, the sets {1, 2, 3} and {a, b, c} are equipotent, while {1, 2} and {a, b, c} are not. For infinite sets, equipotence remains a meaningful notion: many infinite sets are equipotent to each other, such as the natural numbers N and the integers Z, or N and the set of even numbers. The cardinality of N is denoted aleph-null (ℵ0).

Key principles include that if there is a bijection between two sets, they are equipotent, and conversely, equipotence guarantees a bijection. The Cantor–Bernstein–Schroeder theorem provides a practical criterion: if there are injections from A to B and from B to A, then A and B are equipotent. The concept underpins the formal study of cardinal numbers and the comparison of set sizes across finite and infinite contexts.