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Table of Contents
Preface xi
1 Mathematics and Mathematical Activity 1
1.1 What Is Mathematics? 1
1.2 Mathematical Research and Problem Solving 2
1.3 An Example of a Mathematical Research Situation 3
1.4 Conjectures and Theorems 5
1.5 Methods of Reasoning 5
1.6 Why Do We Need Proofs? 7
1.7 Mathematical Writing 8
1.8 Reading a Mathematics Textbook 9
Chapter 1 Problems 11
2 Sets, Numbers, and Axioms 15
2.1 Sets and Numbers from an Intuitive Perspective 15
2.2 Set Equality and Set Inclusion 23
2.3 Venn Diagrams and Set Operations 31
2.4 Undefined Notions and Axioms of Set Theory 45
2.5 Axioms for the Real Numbers 50
Chapter 2 Problems 56
3 Elementary Logic 69
3.1 Statements and Truth 69
3.2 Truth Tables and Statement Forms 80
3.3 Logical Equivalence 86
3.4 Arguments and Validity 91
3.5 Statements Involving Quantifiers 98
Chapter 3 Problems 106
4 Planning and Writing Proofs 117
4.1 The Proof-Writing Context 117
4.2 Proving an If… Then Statement 122
4.3 Proving a For All Statement 129
4.4 The Know/Show Approach to Developing Proofs 136
4.5 Existence and Uniqueness 144
4.6 The Role of Definitions in Creating Proofs 153
4.7 Proving and Expressing a Mathematical Equivalence 160
4.8 Indirect Methods of Proof 168
4.9 Proofs Involving Or. 173
4.10 A Mathematical Research Situation 177
Chapter 4 Problems 183
5 Relations and Functions 199
5.1 Relations 199
5.2 Equivalence Relations and Partitions 207
5.3 Functions 217
5.4 One-to-One Functions, Onto Functions, and Bijections 225
5.5 Inverse Relations and Inverse Functions 235
Chapter 5 Problems 239
6 The Natural Numbers, Induction, and Counting 255
6.1 Axioms for the Natural Numbers 255
6.2 Proof by Induction 258
6.3 Recursive Definition and Strong Induction 270
6.4 Elementary Number Theory 276
6.5 Some Elementary Counting Methods 286
Chapter 6 Problems 298
7 Further Mathematical Explorations 311
7.1 Exploring Graph Theory 311
7.2 Exploring Groups 323
7.3 Exploring Set Cardinality 337
Index 347