equinumerosity

Equinumerosity is a mathematical concept describing when two sets have the same size. Two sets A and B are equinumerous if there exists a bijection between A and B. Equivalently, their cardinalities are equal, written |A| = |B|. This notion is central to the study of cardinality in set theory, where size is defined by the existence of a one-to-one correspondence rather than by counting elements directly.

As an equivalence relation, equinumerosity is reflexive, symmetric, and transitive. Every set is equinumerous with itself, and if A is equinumerous with B and B with C, then A is equinumerous with C. This partitions the class of all sets into equivalence classes, each class representing a cardinal number. Finite sets correspond to natural numbers, while infinite sets correspond to infinite cardinals.

Examples help illustrate the concept. Finite sets A = {1,2,3} and B = {a,b,c} are equinumerous, since a bijection exists between them. An important infinite example is that the set of natural numbers N is equinumerous with the set of even numbers 2N, via the bijection n -> 2n. More generally, many infinite sets can be shown equinumerous by explicit bijections; in contrast, sets with different cardinalities are not equinumerous.

A fundamental result related to equinumerosity is the Cantor–Bernstein–Schröder theorem: if there are injections from A to B and from B to A, then A and B are equinumerous. This underpins the idea that size comparisons can be completed even without constructing a bijection directly.