DEFINITIONS RELATED TO CARDINALITY OF SETS ------------------------------------------ A set A is *finite* if there is a natural number n and a bijection f: {1,...,n} -> A. The number n is called the *cardinality* of A, and we write |A| = n. A set is *infinite* if it is not finite. A set A is *countably infinite* if there is a bijection from N (the set of natural numbers) to A. A set A is *countable* if it is finite or countably infinite. A set A is *uncountable* if it is not countable. Sets A and B have the same cardinality, denoted |A| = |B|, if there exists a bijection from A to B. Set B is at least as large as A, denoted |A| <= |B|, if there exists a one-to-one function from A to B. Set B is larger than A, denoted |A| < |B|, if |A| <= B and not |A| = |B|. The cardinality of the natural numbers is written Aleph not: |N| = Aleph_0 The cardinality of the reals is written c: |R| = c.