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1 THE ELEMENTS OF THE THEORY OF SETS 1

1．1 Introduction 1

1．2 The Concept of Set 3

1．3 Constants 4

1．4 Variables and Equality 7

1．5 Some Basic Notations and Definitions 10

1．6 Subsets;Equality of Sets;The Empty Set 12

1．7 The First Theorem 17

1．8 A(Very)Brief Section on Logic 18

1．9 The Algebra of Sets 25

1．10 Remarks on Notation and Other Matters 34

1．11 Some Special Sets 38

1．12 Ordered Pairs 43

1．13 Cartesian Products,Relations 46

1．14 Functions(or Mappings) 49

1．15 Equivalence Relations and Partitions 63

1．16 Mathematical Systems 72

2 THE NATURAL NUMBERS 74

2．1 The Definition of the Natural Numbers 74

2．2 The Ordering of the Natural Numbers 89

2．3 Counting 98

2．4 Finite Sets 101

2．5 Addition and Multiplication 106

2．6 The Relations between Order,Addition and Multiplication 112

2．7 The Principle of Finite Induction,Again 115

2．8 Sequences 117

2．9 Recursive Definitions 120

3 THE INTEGERS AND THE RATIONAL NUMBERS 132

3．1 Introduction 132

3．2 Definition and Properties of the Integers 133

3．3 Number-Theoretic Properties of the Integers:Generalized Operations 147

3．4 The Rational Numbers 157

3．5 The Arithmetic of the Rational Numbers 161

3．6 Conclusion:Integral Domains and Quotient Fields 173

4 THE REAL NUMBERS 177

4．1 The Mysterious √2 177

4．2 The Arithmetic of Sequences 180

4．3 Cantor Sequences 187

4．4 Null Sequences 194

4．5 The Real Numbers 199

5 THE DEEPER STUDY OF THE REAL NUMBERS 214

5．1 Ordered Fields 214

5．2 Relations between Ordered Fields and R,the Field of Rational Numbers 221

5．3 The Completeness of the Real Numbers 227

5．4 Roots of Real Numbers 239

5．5 More Theorems on Ordered and Complete,Ordered Fields 244

5．6 The Isomorphism of Complete,Ordered Fields 249

5．7 The Complex Numbers 253