Discrete Mathematics Demystified

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CHAPTER 13 Series 13.1 Fundamental Ideas The idea of adding together finitely many numbers is a simple and familiar one. We are so used to the associative and commutative properties of finite sums that we tend not to think about them. Now we shall learn to add together infinitely many numbers, and we will have to be a bit more careful. As Example 13.1 suggests, contradictions can arise if we do not establish certain ground rules. EXAMPLE 13.1 Give an example of an infinite summation process in which the ordinary rules of arithmetic appear to fail. Solution: Consider the infinite sum If we group the terms as 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 +··· (1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) +···
258 Discrete Mathematics Demystified then the sum seems to be 0 + 0 + 0 + 0 +··· which ought to be 0. On the other hand, if we group the terms as then the sum seems to be 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) +··· 1 + 0 + 0 + 0 +··· which ought to be 1. Apparently the associative law of addition fails for this infinite sum. It turns out that, while the associative law is valid for finite sums, it is not valid in general for infinite sums. In order to make sense of this and other addition operations, we must first consider precisely what it means to “add” infinitely many numbers. We begin by recalling the following notation from Sec. 6.4: Definition 13.1 The symbol N c j j=M denotes the sum of the numbers cM, cM+1,...,cN−1, cN . In other words N c j = cM + cM+1 +···+cN−1 + cN j=M EXAMPLE 13.2 If c j = 2 − j then calculate 4 j=1 c j and 7 j=3 c j. Solution: We have 4 j=1 c j = 2 −1 + 2 −2 + 2 −3 + 2 −4 = 15 16
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Discrete Mathematics Demystified
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Discrete Mathematics Demystified St
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To the memory of J. W. T. Youngs.
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CONTENTS Preface xiii CHAPTER 1 Log
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Contents ix CHAPTER 7 Matrices 135
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Contents xi Final Exam 301 Solution
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PREFACE In today’s world, analyti
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CHAPTER 1 Strictly speaking, our ap
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CHAPTER 1 Logic 3 The point of view
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CHAPTER 1 Logic 5 A B A∨ B T T T
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CHAPTER 1 Logic 7 In the next two s
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CHAPTER 1 Logic 9 the hypothesis (2
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CHAPTER 1 Logic 11 is the line in w
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CHAPTER 1 Logic 13 is the statement
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CHAPTER 1 Logic 15 and EXAMPLE 1.22
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CHAPTER 1 Logic 17 EXAMPLE 1.25 Loo
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CHAPTER 1 Logic 19 We conclude by n
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CHAPTER 1 Logic 21 6. For each of t
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CHAPTER 2 Methods of Mathematical P
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CHAPTER 3 Set Theory 3.1 Rudiments
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CHAPTER 3 Set Theory 43 Definition
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CHAPTER 3 Set Theory 45 EXAMPLE 3.8
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CHAPTER 3 Set Theory 47 A Figure 3.
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CHAPTER 3 Set Theory 49 Exercises 1
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CHAPTER 4 Functions and Relations 4
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CHAPTER 5 Number Systems 5.1 Prelim
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CHAPTER 5 Number Systems 69 P3 asse
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CHAPTER 5 Number Systems 71 Proposi
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CHAPTER 5 Number Systems 73 on fait
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CHAPTER 5 Number Systems 75 Adding
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CHAPTER 5 Number Systems 77 answer;
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CHAPTER 5 Number Systems 79 In an e
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CHAPTER 5 Number Systems 81 the quo
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CHAPTER 5 Number Systems 83 Theorem
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CHAPTER 5 Number Systems 85 Likewis
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CHAPTER 5 Number Systems 87 EXAMPLE
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CHAPTER 5 Number Systems 89 Similar
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CHAPTER 5 Number Systems 91 5.5.1 C
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CHAPTER 5 Number Systems 93 Notice
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CHAPTER 5 Number Systems 95 maximal
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CHAPTER 5 Number Systems 97 Remark
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CHAPTER 5 Number Systems 99 Theorem
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CHAPTER 5 Number Systems 101 the ab
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CHAPTER 5 Number Systems 103 4. Exp
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CHAPTER 6 Counting Arguments Althou
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CHAPTER 6 Counting Arguments 107 Fi
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CHAPTER 6 Counting Arguments 109 A
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CHAPTER 6 Counting Arguments 111 A
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CHAPTER 6 Counting Arguments 113 No
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CHAPTER 6 Counting Arguments 115 We
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CHAPTER 6 Counting Arguments 117 (o
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CHAPTER 6 Counting Arguments 119 he
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CHAPTER 6 Counting Arguments 121 an
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CHAPTER 6 Counting Arguments 123 It
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CHAPTER 6 Counting Arguments 125 ca
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CHAPTER 6 Counting Arguments 127 EX
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CHAPTER 6 Counting Arguments 129 k
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CHAPTER 6 Counting Arguments 131 Fi
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CHAPTER 6 Counting Arguments 133 2.
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CHAPTER 7 Matrices 7.1 What Is a Ma
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CHAPTER 7 Matrices 137 EXAMPLE 7.2
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CHAPTER 7 Matrices 139 and Calculat
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CHAPTER 7 Matrices 141 are all poin
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CHAPTER 7 Matrices 143 We encourage
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CHAPTER 7 Matrices 145 Translating
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CHAPTER 7 Matrices 147 If we guess
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CHAPTER 7 Matrices 149 Solution: We
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CHAPTER 7 Matrices 151 Finally we m
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CHAPTER 7 Matrices 153 7.5 Markov C
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CHAPTER 7 Matrices 155 Thus we disc
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CHAPTER 7 Matrices 157 Figure 7.2 A
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CHAPTER 7 Matrices 159 Figure 7.4 T
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CHAPTER 7 Matrices 161 f (0, 1, 1)
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CHAPTER 8 Graph Theory 8.1 Introduc
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CHAPTER 8 Graph Theory 165 in the e
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CHAPTER 8 Graph Theory 167 Figure 8
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CHAPTER 8 Graph Theory 169 8.3 Appl
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CHAPTER 8 Graph Theory 173 Not a co
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CHAPTER 8 Graph Theory 177 number o
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CHAPTER 8 Graph Theory 179 $65 D A
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CHAPTER 8 Graph Theory 181 Exercise
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CHAPTER 9 Number Theory 9.1 Divisib
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CHAPTER 9 Number Theory 185 k—the
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CHAPTER 9 Number Theory 187 then Al
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CHAPTER 9 Number Theory 189 Axiom 3
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CHAPTER 9 Number Theory 191 EXAMPLE
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CHAPTER 9 Number Theory 193 collect
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CHAPTER 9 Number Theory 195 are the
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CHAPTER 9 Number Theory 197 We may
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CHAPTER 10 Cryptography 10.1 Backgr
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CHAPTER 10 Cryptography 201 1 1 0 1
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CHAPTER 10 Cryptography 203 When wa
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CHAPTER 10 Cryptography 205 Thus if
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Final Exam 307 (c) An irrational nu
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Final Exam 309 (d) A useful device
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Final Exam 311 (c) 0.00004 (d) 0.02
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Final Exam 313 (c) x = 5, y = 3, z
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Final Exam 315 (c) 3 (d) 5 (e) 1 63
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Final Exam 317 (c) Z/4Z (d) R (e) Q
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Final Exam 319 82. The boolean expr
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Final Exam 321 (d) Displays. (e) Ra
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Final Exam 323 Solutions 1. (b) 21.
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Solutions to Exercises This book ha
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Solutions to Exercises 327 Chapter
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Solutions to Exercises 329 10. If o
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Solutions to Exercises 331 and ther
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Solutions to Exercises 333 5. -6 -
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Solutions to Exercises 335 among th
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Solutions to Exercises 337 5. −1
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Solutions to Exercises 339 Figure 8
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Solutions to Exercises 341 Givena2
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Solutions to Exercises 343 3. We no
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Solutions to Exercises 345 Chapter
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Bibliography [ADA] J. F. Adams, On
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A Abel, Niels Henrik, 188 Abelian g
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INDEX 351 relations, 65 sequences,
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INDEX 353 Partitioning, 54 Pascal,
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Discrete Mathematics Demystified

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