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The size of a set, understood up to bijection
A cardinality is the size of a set, understood up to bijection: two sets and have the same cardinality if there exists a bijective function . The cardinality of is often denoted .
Cardinality comparisons can be expressed using injections and surjections between sets. A set is countable when its cardinality is at most that of the natural numbers.
In axiomatic set theory, set and the membership relation are primitive notions. The axiom of extensionality determines equality by membership: for sets ,
A bijective function is a function that is both injective and surjective.
A countable set is a set for which there exists an injective function , where is the set of natural numbers. Equivalently, is countable if it is finite or there exists a bijective function between and .
The empty set, denoted , is the set with no elements:
A power set of a set is the set