Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop. 2017 Spis treści Preface xiii I Fundamentals of Abstract Mathematics 1 1 Basic Notions 3 1.1 A First Look at Some Familiar Number Systems 3 1.1.1 Integers and natural numbers 3 1.1.2 Rational numbers and real numbers 7 1.2 Inequalities 9 1.3 A First Look at Sets and Functions 12 1.3.1 Sets, elements, and subsets 12 1.3.2 Operations with sets 13 1.3.3 Special subsets of : intervals 13 1.3.4 Functions 14 1.4 Problems 15 2 Mathematical Induction 17 2.1 First Examples 17 2.1.1 Defining sequences through a formula for the n-th term 18 2.1.2 Defining sequences recursively 20 2.2 First Programs 23 2.3 First Proofs: Mathematical Induction 25 2.4 Strong Induction 27 2.5 The Well-Ordering Principle and Induction 28 2.6 Problems 29 3 Basic Logic and Proof Techniques 33 3.1 Logical Statements and Truth Tables 33 3.1.1 Statements and their negations 33 3.1.2 Combining statements 34 3.1.3 Implications 35 3.2 Quantified Statements and Their Negations 37 3.2.1 Writing implications as quantified statements 39 3.3 Proof Techniques 40 3.3.1 Direct proof 40 3.3.2 Proof by contradiction 41 3.3.3 Proof by contraposition 41 3.3.4 The art of the counterexample 42
3.4 Problems 43 4 Sets, Relations, and Functions 47 4.1 Sets 47 4.2 Relations 49 4.2.1 The definition 49 4.2.2 Order relations 50 4.2.3 Equivalence relations 50 4.3 Functions 52 4.3.1 Images and pre-images 52 4.3.2 Injections, surjections, and bijections 53 4.3.3 Compositions of functions 54 4.3.4 Inverse functions 56 4.4 Problems 56 5 Elementary Discrete Mathematics 61 5.1 Basic Principles of Combinatorics 61 5.1.1 The Addition and Multiplication Principles 61 5.1.2 Permutations and combinations 64 5.1.3 Combinatorial identities 65 5.2 Linear Recurrence Relations 69 5.2.1 An example 69 5.2.2 General results 71 5.3 Analysis of Algorithms 72 5.3.1 Some simple algorithms 72 5.3.2 O, Ω, and Θ notation 75 5.3.3 Analysis of the binary search algorithm 76 5.4 Problems 79 6 Number Systems; Algebraic Structures 81 6.1 Representations of Natural Numbers 81 6.1.1 Developing an algorithm to convert a number from base 10 to base 2 83 6.1.2 Proof of the existence and uniqueness of the base b representation of an element of 86 6.2 Integers and Divisibility 89 6.3 Modular Arithmetic 94 6.3.1 Definition of congruence and basic properties 94 6.3.2 Congruence classes 96 6.3.3 Operations on congruence classes 97 6.4 The Rational Numbers 99 6.5 Algebraic Structures 102 6.5.1 Binary operations 103 6.5.2 Groups 104 6.5.3 Rings and fields 107
6.6 Problems 110 7 Cardinality 113 7.1 The Definition 113 7.2 Finite Sets Revisited 114 7.3 Countably Infinite Sets 116 7.4 Uncountable Sets 119 7.5 Problems 121 II Foundations of Analysis 123 8 Sequences of Real Numbers 127 8.1 The Limit of a Sequence 127 8.1.1 Numerical and graphical exploration 127 8.1.2 The precise definition of a limit 130 8.2 Properties of Limits 132 8.3 Cauchy Sequences 134 8.3.1 Showing that a sequence is Cauchy 136 8.3.2 Showing that a sequence is divergent 136 8.3.3 Properties of Cauchy sequences 137 8.4 Problems 138 9 A Closer Look at the Real Number System 141 9.1 Rasa Complete Ordered Field 141 9.1.1 Completeness 141 9.1.2 Why is not complete 147 9.1.3 Algorithms for approximating 2 148 9.2 Construction of 150 9.2.1 An equivalence relation on Cauchy sequences of rational numbers 151 9.2.2 Operations on 152 9.2.3 Verifying the field axioms 153 9.2.4 Defining order 155 9.2.5 Sequences of real numbers and completeness 158 9.3 Problems 160 10 Series, Part 1 163 10.1 Basic Notions 163 10.1.1 Definitions 163 10.1.2 Exploring the sequence of partial sums graphically and numerically 165 10.1.3 Basic properties of convergent series 168 10.1.4 Series that diverge slowly: The harmonic series 169 10.2 Infinite Geometric Series 172 10.3 Tests for Convergence of Series 173
10.4 Representations of Real Numbers 180 10.4.1 Base 10 representation 181 10.4.2 Base 10 representations of rational numbers 183 10.4.3 Representations in other bases 185 10.5 Problems 186 11 The Structure of the Real Line 189 11.1 Basic Notions from Topology 190 11.1.1 Open and closed sets 190 11.1.2 Accumulation points of sets 193 11.2 Compact sets 195 11.2.1 Subsequences and limit points 195 11.2.2 First definition of compactness 196 11.2.3 The Heine-Borel property 199 11.3 A First Glimpse at the Notion of Measure 201 11.3.1 Measuring intervals 201 11.3.2 Measure zero 205 11.3.3 The Cantor set 207 11.4 Problems 208 12 Continuous Functions 211 12.1 Sequential Continuity 211 12.1.1 Exploring sequential continuity graphically and numerically 211 12.1.2 Proving that a function is continuous 213 12.1.3 Proving that a function is discontinuous 214 12.1.4 First results 215 12.2 Related Notions 216 12.2.1 The ε-δ condition 216 12.2.2 Uniform continuity 218 12.2.3 The limit of a function 221 12.3 Important Theorems 223 12.3.1 The Intermediate Value Theorem 223 12.3.2 Developing a root-finding algorithm from the proof of the IVT 224 12.3.3 Continuous functions on compact intervals 225 12.4 Problems 226 13 Differentiation 229 13.1 Definition and First Examples 229 13.2 Differentiation Rules 232 13.3 Applications of the Derivative 236 13.4 Problems 239 14 Series, Part 2 241 14.1 Absolute and Conditional Convergence 241 14.1.1 The first example 242
14.1.2 Summation by Parts and the Alternating Series Test 244 14.1.3 Basic facts about conditionally convergent series 249 14.2 Rearrangements 251 14.2.1 Rearrangements and non-negative series 251 14.2.2 Using Python to explore the alternating harmonic series 252 14.2.3 A general theorem 254 14.3 Problems 257 A A Very Short Course on Python 259 A.1 Getting Started 259 A.1.1 Why Python? 259 A.1.2 Python versions 2 and 3 260 A.2 Installation and Requirements 260 A.2.1 Integrated Development Environments (IDEs) 260 A.3 Python Basics 260 A.3.1 Exploring in the Python console 260 A.3.2 Your first programs 262 A.3.3 Good programming practice 263 A.3.4 Lists and strings 265 A.3.5 if else structures and comparison operators 266 A.3.6 Loop structures 268 A.4 Functions 270 A.5 Recursion 273 Index 275 oprac. BPK