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Chapter 2 The Basic Concepts of Set Theory © 2008 Pearson Addison-Wesley. All rights reserved

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-2 Chapter 2: The Basic Concepts of Set Theory 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Surveys and Cardinal Numbers 2.5Infinite Sets and Their Cardinalities

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-3 Chapter 1 Section 2-5 Infinite Sets and Their Cardinalities

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-4 Infinite Sets and Their Cardinalities One-to-One Correspondence and Equivalent Sets The Cardinal Number (Aleph-Null) Infinite Sets Sets That Are Not Countable

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-5 One-to-One Correspondence and Equivalent Sets A one-to-one correspondence between two sets is a pairing where each element of one set is paired with exactly one element of the second set and each element of the second set is paired with exactly one element of the first set.

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-6 For sets {a, b, c, d} and {3, 7, 9, 11} a pairing to demonstrate one-to-one correspondence could be {a, b, c, d} {3, 7, 9, 11} Example: One-to-One Correspondence

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-7 Equivalent Sets Two sets, A and B, which may be put in a one-to-one correspondence are said to be equivalent, written A ~ B.

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-8 The Cardinal Number The basic set used in discussing infinite sets is the set of counting numbers, {1, 2, 3, …}. The set of counting numbers is said to have the infinite cardinal number (aleph-null).

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-9 To show that another set has cardinal number we show that it is equivalent to the set of counting numbers. {1, 2, 3, 4, …, n, …} {2, 4, 6, 8, …,2n, …} Example: Showing That {2, 4, 6, 8,…} Has Cardinal Number

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-10 Infinite Sets A set is infinite if it can be placed in a one- to-one correspondence with a proper subset of itself. The whole numbers, integers, and rational numbers have cardinal number

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-11 Countable Sets A set is countable if it is finite or if it has cardinal number

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-12 Sets That Are Not Countable The real numbers and irrational numbers are not countable and are said to have cardinal number c (for continuum).

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© 2008 Pearson Addison-Wesley. All rights reserved 2-5-13 Cardinal Numbers of Infinite Sets Infinite SetCardinal Number Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers c Real Numbers c

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