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34 Further Ideas in Elementary Set Theory Exercises 47 49 51 51 53 56 59 63 65 67 67 68 73 79 86 94 96 101 102 105 105 108 110 113 118 121 124 127 130 132
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Functions and Relations 41 A Word About Number Systems 42 Relations and Functions 43 Functions 44 Combining Functions 45 Types of Functions Exercises Number Systems 51 Preliminary Remarks 52 The Natural Number System 53 The Integers 54 The Rational Numbers 55 The Real Number System 56 The Nonstandard Real Number System 57 The Complex Numbers 58 The Quaternions, the Cayley Numbers, and Beyond Exercises Counting Arguments 61 The Pigeonhole Principle 62 Orders and Permutations 63 Choosing and the Binomial Coef cients 64 Other Counting Arguments 65 Generating Functions 66 A Few Words About Recursion Relations 67 Probability 68 Pascal s Triangle 69 Ramsey Theory Exercises
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CHAPTER 7 Matrices 71 What Is a Matrix 72 Fundamental Operations on Matrices 73 Gaussian Elimination 74 The Inverse of a Matrix 75 Markov Chains 76 Linear Programming Exercises Graph Theory 81 Introduction 82 Fundamental Ideas of Graph Theory 83 Application to the K onigsberg Bridge Problem 84 Coloring Problems 85 The Traveling Salesman Problem Exercises Number Theory 91 Divisibility 92 Primes 93 Modular Arithmetic 94 The Concept of a Group 95 Some Theorems of Fermat Exercises Cryptography 101 Background on Alan Turing 102 The Turing Machine 103 More on the Life of Alan Turing 104 What Is Cryptography 105 Encryption by Way of Af ne Transformations 106 Digraph Transformations
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ix 135 135 136 139 145 153 156 161 163 163 165 169 172 178 181 183 183 185 186 187 196 197 199 199 200 202 203 209 216
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107 RSA Encryption Exercises 221 233 235 235 236 238 239 241 243 243 244 250 250 253 254 256 257 257 260 263 265 266 268 269 270 273 279 283 288 291 294 298
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Boolean Algebra 111 Description of Boolean Algebra 112 Axioms of Boolean Algebra 113 Theorems in Boolean Algebra 114 Illustration of the Use of Boolean Logic Exercises Sequences 121 Introductory Remarks 122 In nite Sequences of Real Numbers 123 The Tail of a Sequence 124 A Basic Theorem 125 The Pinching Theorem 126 Some Special Sequences Exercises Series 131 Fundamental Ideas 132 Some Examples 133 The Harmonic Series 134 Series of Powers 135 Repeating Decimals 136 An Application 137 A Basic Test for Convergence 138 Basic Properties of Series 139 Geometric Series 1310 Convergence of p-Series 1311 The Comparison Test 1312 A Test for Divergence 1313 The Ratio Test 1314 The Root Test Exercises
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Contents
Final Exam Solutions to Exercises Bibliography Index
xi 301 325 347 349
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PREFACE
In today s world, analytical thinking is a critical part of any solid education An important segment of this kind of reasoning one that cuts across many disciplines is discrete mathematics Discrete math concerns counting, probability, (sophisticated forms of) addition, and limit processes over discrete sets Combinatorics, graph theory, the idea of function, recurrence relations, permutations, and set theory are all part of discrete math Sequences and series are among the most important applications of these ideas Discrete mathematics is an essential part of the foundations of (theoretical) computer science, statistics, probability theory, and algebra The ideas come up repeatedly in different parts of calculus Many would argue that discrete math is the most important component of all modern mathematical thought Most basic math courses (at the freshman and sophomore level) are oriented toward problem-solving Students can rely heavily on the provided examples as a crutch to learn the basic techniques and pass the exams Discrete mathematics is, by contrast, rather theoretical It involves proofs and ideas and abstraction Freshman and sophomores in college these days have little experience with theory or with abstract thinking They simply are not intellectually prepared for such material Steven G Krantz is an award-winning teacher, author of the book How to Teach Mathematics He knows how to present mathematical ideas in a concrete fashion that students can absorb and master in a comfortable fashion He can explain even abstract concepts in a hands-on fashion, making the learning process natural and uid Examples can be made tactile and real, thus helping students to nesse abstract technicalities This book will serve as an ideal supplement to any standard text It will help students over the traditional hump that the rst theoretical math course constitutes It will make the course palatable Krantz has already authored two successful Demysti ed books The good news is that discrete math, particularly sequences and series, can be illustrated with concrete examples from the real world They can be made to be realistic and approachable Thus the rather dif cult set of ideas can be made accessible to a broad audience of students For today s audience consisting not
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