Mathematical Thinking and Writing (eBook)

Cover 1
Contents 8
Why Read This Book? 14
Preface 16
Chapter 0. Notation and Assumptions 20
0.1 Set Terminology and Notation 20
0.2 Assumptions 24
Part I: Foundations of Logic and Proof Writing 30
Chapter 1. Logic 32
1.1 Introduction to Logic 32
1.2 If-Then Statements 39
1.3 Universal and Existential Quantifiers 46
1.4 Negations of Statements 50
Chapter 2. Beginner-Level Proofs 57
2.1 Proofs Involving Sets 57
2.2 Indexed Families of Sets 66
2.3 Algebraic and Ordering Properties of R 72
2.4 The Principle of Mathematical Induction 80
2.5 Equivalence Relations: The Idea of Equality 87
2.6 Equality, Addition, and Multiplication inQ 95
2.7 The Division Algorithm and Divisibility 98
2.8 Roots and irrational numbers 104
2.9 Relations In General 109
Chapter 3. Functions 116
3.1 Definitions and Terminology 116
3.2 Composition and Inverse Functions 125
3.3 Cardinality of Sets 129
3.4 Counting Methods and the Binomial Theorem 137
Part II: Basic Priniciples of Analysis 150
Chapter 4. The Real Numbers 152
4.1 The Least Upper Bound Axiom 153
4.2 Sets in R 159
4.3 Limit Points and Closure of Sets 162
4.4 Compactness 165
4.5 Sequences in R 168
4.6 Convergence of Sequences 172
4.7 The Nested Interval Property 179
4.8 Cauchy Sequences 184
Chapter 5. Functions of a Real Variable 189
5.1 Bounded and Monotone Functions 189
5.2 Limits and Their Basic Properties 192
5.3 More on Limits 199
5.4 Limits Involving Infinity 201
5.5 Continuity 206
5.6 Implications of Continuity 214
5.7 Uniform Continuity 219
Part III: Basic Principles of Alegbra 224
Chapter 6. Groups 226
6.1 Introduction to Groups 226
6.2 Generated and Cyclic Subgroups 234
6.3 Integers Modulo n and Quotient Groups 239
6.4 Permutation Groups and Normal Subgroups 246
6.5 Group Morphisms 255
Chapter 7. Rings 262
7.1 Rings and Subrings 262
7.2 Ring Properties and Fields 268
7.3 Ring Extensions 275
7.4 Ideals 279
7.5 Integral Domains 286
7.6 UFDs and PIDs 292
7.7 Euclidean Domains 298
7.8 Ring Morphisms 306
7.9 Quotient Rings 310
Index 318