CSE541 CARDINALITIES OF SETS BASIC DEFINITIONS AND FACTS 1¡1;onto Cardinality de¯nition Sets A and B have the same cardinality i® 9f( f : A ¡! B): Cardinality notations If sets A and B have the same cardinality we denote it as: jAj = jBj or cardA = cardB, or A » B. We aslo say that A and B are equipotent. Cardinality We put the above notations together in one de¯nition: 1¡1;onto jAj = jBj or cardA = cardB, or A » B i® 9f( f : A ¡! B): 1¡1;onto Finite A set A is ¯nite i® 9n 2 N 9f (f : f0; 1; 2; :::; n ¡ 1g ¡! A), i.e. we say: a set A is ¯nite i® 9n 2 N(jAj = n). In¯nite A set A is in¯nite i® A is NOT ¯nite. Cardinality Aleph zero @0 (Aleph zero) is a cardinality of N (Natural numbers). A set A has a cardinality @0 (jAj = @0) i® A » N, (or jAj = jNj, or cardA = cardN). Countable A set A is countable i® A is ¯nite or jAj = @0. In¯nitely countable A set A is in¯nitely countable i® jAj = @0. Uncountable A set A is uncountable i® A is NOT countable, Fact A set A is uncountable i® A is in¯nite and jAj 6= @0. Cardinality Continuum C (Continuum ) is a cardinality of Real Numbers, i.e. C = jRj. Sets of cardinality continuum. A set A has a cardinality C, jAj = C i® jAj = jRj, or cardA = cardR). Cardinality A · Cardinality B jAj · jBj i® A » C and C ⊆ B. Fact If A ⊆ B then jAj · jBj. Denote jAj = N and jBj = M. We write it symbolically: N·M when jAj · jBj. Cardinality A < Cardinality B jAj < jBj i® jAj · jBj and jAj 6= jBj. Denote jAj = N and jBj = M. We write it symbolically: N < M when jAj < jBj. Cantor Theorem For any set A, jAj < jP(A)j. Cantor-Berstein Theorem For any sets A; B, If jAj · jBj and jBj · jAj, then jAj = jBj. Denote jAj = N and jBj = M. We write it symbolically: If N·M and M·N , then N = M. ARITHMETIC OF CARDINAL NUMBERS Sum ( N + M) We de¯ne: N + M = jA [ Bj, where A; B are such that jAj = N , jBj = M and A \ B = ;. 1 Multiplication ( N ¢ M) We de¯ne: N ¢ M = jA £ Bj, where A; B are such that jAj = N , jBj = M. Power ( MN ) MN = cardff : f : A ¡! Bg, where A; B are such that jAj = N , jBj = M. I.e. MN is the cardinality of all functions that map a set A (of cardinality N ) into a set B (of cardinality M). Power ( 2N ) 2N = cardff : f : A ¡! f0; 1g g, where jAj = N . 2N = cardP(A), where jAj = N . Power Rules N P+T = N P ¢ N T .(N P )T = N P¢T . ARITHMETIC OF n, @0, C Union 1 @0 + @0 = @0. Union of two in¯nitely countable sets is an in¯nitely countable set. Union 2 @0 + n = @0. Union of a ¯nite ( cardinality n) and in¯nitely countable set is an in¯nitely countable set. Union 3 @0 + C = C. Union of an in¯nitely countable set and a set of the same cardinality as Real numbers has the same cardinality as the set of Real numbers. Union 4 C + C = C. Union of two sets of cardinality the same as Real numbers has the same cardinality as the set of Real numbers. Cartesian Product 1 @0 ¢ @0 = @0. Cartesian Product of two in¯nitely countable sets is an in¯nitely countable set. Cartesian Product 2 n ¢ @0 = @0. Cartesian Product of a ¯nite set and an in¯nitely countable set is an in¯nitely countable set. Cartesian Product 3 @0 ¢ C = C. Cartesian Product of an in¯nitely countable set and a set of the same cardinality as Real numbers has the same cardinality as the set of Real numbers. Cartesian Product 4 C ¢ C = C. Cartesian Product of two sets of cardinality the same as Real numbers has the same cardinality as the set of Real numbers. Power 1 2@0 = C. The set of all subsets of Natural numbers (or any set equipotent with natural numbers) has the same cardinality as the set of Real numbers. @0 Power 2 @0 = C. There are Cof all functions that map N into N. There are C sequences (all sequences) that can be form out of an in¯nitely countable set. @0 @0 = ff : f : N ¡! Ng = C. Power 3 CC = 2C. There are 2C of all functions that map R into R. The set of all real functions of one variable has the same cardinality as the set of all subsets of Real numbers. 2 Inequalities n < @0 < C. ¤ ¤ Theorem If A is a ¯nite set, A is the set of all ¯nite sequences formed out of A, then A has @0 elements. ¤ Shortly: If jAj = n, then jA j = @0. 3.