Transition to Abstract Mathematics (eBook)

Front Cover 1
A Transition to Abstract Mathematics 4
Copyright Page 5
Table of Contents 8
Why Read This Book? 14
Preface 16
Preface to the First Edition 18
Acknowledgments 22
Chapter 0. Notation and Assumptions 24
0.1 Set Terminology and Notation 24
0.2 Assumptions about the Real Numbers 26
0.2.1 Basic Algebraic Properties 26
0.2.2 Ordering Properties 28
0.2.3 Other Assumptions 30
Part 1: Foundations of Logic and ProofWriting 32
Chapter 1. Language and Mathematics 34
1.1 Introduction to Logic 34
1.1.1 Statements 34
1.1.2 Negation of a Statement 36
1.1.3 Combining Statements with AND 36
1.1.4 Combining Statements with OR 37
1.1.5 Logical Equivalence 39
1.1.6 Tautologies and Contradictions 41
1.2 If-Then Statements 41
1.2.1 If-Then Statements Defined 41
1.2.2 Variations on p . q 44
1.2.3 Logical Equivalence and Tautologies 46
1.3 Universal and Existential Quantifiers 50
1.3.1 The Universal Quantifier 51
1.3.2 The Existential Quantifier 52
1.3.3 Unique Existence 55
1.4 Negations of Statements 56
1.4.1 Negations of AND and OR Statements 56
1.4.2 Negations of If-Then Statements 57
1.4.3 Negations of Statements with the Universal Quantifier 59
1.4.4 Negations of Statements with the Existential Quantifier 60
1.5 How We Write Proofs 63
1.5.1 Direct Proof 63
1.5.2 Proof by Contrapositive 64
1.5.3 Proving a Logically Equivalent Statement 64
1.5.4 Proof by Contradiction 65
1.5.5 Disproving a Statement 65
Chapter 2. Properties of Real Numbers 68
2.1 Basic Algebraic Properties of Real Numbers 68
2.1.1 Properties of Addition 69
2.1.2 Properties of Multiplication 72
2.2 Ordering Properties of the Real Numbers 74
2.3 Absolute Value 76
2.4 The Division Algorithm 79
2.5 Divisibility and Prime Numbers 82
Chapter 3. Sets and Their Properties 86
3.1 Set Terminology 86
3.2 Proving Basic Set Properties 90
3.3 Families of Sets 94
3.4 The Principle of Mathematical Induction 101
3.5 Variations of the PMI 108
3.6 Equivalence Relations 114
3.7 Equivalence Classes and Partitions 120
3.8 Building the Rational Numbers 125
3.8.1 Defining Rational Equality 126
3.8.2 Rational Addition and Multiplication 127
3.9 Roots of Real Numbers 129
3.10 Irrational Numbers 130
3.11 Relations in General 134
Chapter 4. Functions 142
4.1 Definition and Examples 142
4.2 One-to-one and Onto Functions 148
4.3 Image and Pre-Image Sets 151
4.4 Composition and Inverse Functions 154
4.4.1 Composition of Functions 155
4.4.2 Inverse Functions 156
4.5 Three Helpful Theorems 158
4.6 Finite Sets 160
4.7 Infinite Sets 162
4.8 Cartesian Products and Cardinality 167
4.8.1 Cartesian Products 167
4.8.2 Functions Between Finite Sets 169
4.8.3 Applications 171
4.9 Combinations and Partitions 174
4.9.1 Combinations 174
4.9.2 Partitioning a Set 175
4.9.3 Applications 176
4.10 The Binomial Theorem 180
Part II: Basic Principles of Analysis 186
Chapter 5: The Real Numbers 188
5.1 The Least Upper Bound Axiom 188
5.1.1 Least Upper Bounds 189
5.1.2 Greatest Lower Bounds 191
5.2 The Archimedean Property 192
5.2.1 Maximum and Minimum of Finite Sets 193
5.3 Open and Closed Sets 195
5.4 Interior, Exterior, Boundary, and Cluster Points 198
5.4.1 Interior, Exterior, and Boundary 198
5.4.2 Cluster Points 199
5.5 Closure of Sets 201
5.6 Compactness 203
Chapter 6. Sequences of Real Numbers 208
6.1 Sequences Defined 208
6.1.1 Monotone Sequences 209
6.1.2 Bounded Sequences 210
6.2 Convergence of Sequences 213
6.2.1 Convergence to a Real Number 213
6.2.2 Convergence to Infinity 219
6.3 The Nested Interval Property 220
6.3.1 From LUB Axiom to NIP 221
6.3.2 The NIP Applied to Subsequences 222
6.3.3 From NIP to LUB Axiom 224
6.4 Cauchy Sequences 225
6.4.1 Convergence of Cauchy Sequences 226
6.4.2 From Completeness to the NIP 228
Chapter 7. Functions of a Real Variable 230
7.1 Bounded and Monotone Functions 230
7.1.1 Bounded Functions 230
7.1.2 Monotone Functions 231
7.2 Limits and Their Basic Properties 233
7.2.1 Definition of Limit 233
7.2.2 Basic Theorems of Limits 236
7.3 More on Limits 240
7.3.1 One-Sided Limits 240
7.3.2 Sequential Limits 241
7.4 Limits Involving Infinity 242
7.4.1 Limits at Infinity 243
7.4.2 Limits of Infinity 245
7.5 Continuity 247
7.5.1 Continuity at a Point 247
7.5.2 Continuity on a Set 251
7.5.3 One-Sided Continuity 253
7.6 Implications of Continuity 254
7.6.1 The Intermediate Value Theorem 254
7.6.2 Continuity and Open Sets 256
7.7 Uniform Continuity 258
7.7.1 Definition and Examples 259
7.7.2 Uniform Continuity and Compact Sets 262
Part III: Basic Principles of Algebra 264
Chapter 8. Groups 266
8.1 Introduction to Groups 266
8.1.1 Basic Characteristics of Algebraic Structures 266
8.1.2 Groups Defined 269
8.1.1 Basic Characteristics of Algebraic Structures 266
8.1.2 Groups Defined 269
8.2 Subgroups 275
8.2.1 Subgroups Defined 275
8.2.2 Generated Subgroups 277
8.2.3 Cyclic Subgroups 278
8.3 Quotient Groups 283
8.3.1 Integers Modulo n 283
8.3.2 Quotient Groups 286
8.3.3 Cosets and Lagrange’s Theorem 290
8.4 Permutation Groups 291
8.4.1 Permutation Groups Defined 291
8.4.2 The Symmetric Group 292
8.4.3 The Alternating Group 294
8.4.4 The Dihedral Group 296
8.5 Normal Subgroups 298
8.6 Group Morphisms 303
Chapter 9. Rings 310
9.1 Rings and Fields 310
9.1.1 Rings Defined 310
9.1.2 Fields Defined 315
9.2 Subrings 316
9.3 Ring Properties 319
9.4 Ring Extensions 324
9.4.1 Adjoining Roots of Ring Elements 324
9.4.2 Polynomial Rings 327
9.4.3 Degree of a Polynomial 328
9.5 Ideals 329
9.6 Generated Ideals 332
9.7 Prime and Maximal Ideals 335
9.8 Integral Domains 337
9.9 Unique Factorization Domains 342
9.10 Principal Ideal Domains 344
9.11 Euclidean Domains 348
9.12 Polynomials over a Field 351
9.13 Polynomials over the Integers 355
9.14 Ring Morphisms 357
9.14.1 Properties of Ring Morphisms 359
9.15 Quotient Rings 362
Index 368