Equipollent (idea)

In mathematics, equipollence is a concept in set theory. Two sets are equipollent iff there is a bijection from one to the other.

The symbol for this is a wavy equals-sign.

Informally, the two sets match each other in size. This is a precursor notion to being able to define the cardinality of a set. Being equipollent constitutes an equivalence relation on the class of sets:

• (reflexive) The identity mapping on any set A to itself is bijective.
• (symmetric) If f is a bijection from A to B then its inverse f^-1 is well-defined and is a bijection from B to A.
• (transitive) If f is bijective from A to B, and g from B to C, then fg is well-defined and is a bijection from A to C.

The statement that A is equipollent to B may also be phrased as A and B have the same cardinality, and a notion of cardinality may be invoked, symbolized |A|, such that you can write that as |A| = |B|. But it requires substantially more work (including the Cantor-Schröder-Bernstein Theorem) to be able to treat this "cardinality" as a cardinal with all its properties.