Discrete Mathematics Demystified

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Contents ix CHAPTER 7 Matrices 135 7.1 What Is a Matrix? 135 7.2 Fundamental Operations on Matrices 136 7.3 Gaussian Elimination 139 7.4 The Inverse of a Matrix 145 7.5 Markov Chains 153 7.6 Linear Programming 156 Exercises 161 CHAPTER 8 Graph Theory 163 8.1 Introduction 163 8.2 Fundamental Ideas of Graph Theory 165 8.3 Application to the Königsberg Bridge Problem 169 8.4 Coloring Problems 172 8.5 The Traveling Salesman Problem 178 Exercises 181 CHAPTER 9 Number Theory 183 9.1 Divisibility 183 9.2 Primes 185 9.3 Modular Arithmetic 186 9.4 The Concept of a Group 187 9.5 Some Theorems of Fermat 196 Exercises 197 CHAPTER 10 Cryptography 199 10.1 Background on Alan Turing 199 10.2 The Turing Machine 200 10.3 More on the Life of Alan Turing 202 10.4 What Is Cryptography? 203 10.5 Encryption by Way of Affine Transformations 209 10.6 Digraph Transformations 216
x Discrete Mathematics Demystified 10.7 RSA Encryption 221 Exercises 233 CHAPTER 11 Boolean Algebra 235 11.1 Description of Boolean Algebra 235 11.2 Axioms of Boolean Algebra 236 11.3 Theorems in Boolean Algebra 238 11.4 Illustration of the Use of Boolean Logic 239 Exercises 241 CHAPTER 12 Sequences 243 12.1 Introductory Remarks 243 12.2 Infinite Sequences of Real Numbers 244 12.3 The Tail of a Sequence 250 12.4 A Basic Theorem 250 12.5 The Pinching Theorem 253 12.6 Some Special Sequences 254 Exercises 256 CHAPTER 13 Series 257 13.1 Fundamental Ideas 257 13.2 Some Examples 260 13.3 The Harmonic Series 263 13.4 Series of Powers 265 13.5 Repeating Decimals 266 13.6 An Application 268 13.7 A Basic Test for Convergence 269 13.8 Basic Properties of Series 270 13.9 Geometric Series 273 13.10 Convergence of p-Series 279 13.11 The Comparison Test 283 13.12 A Test for Divergence 288 13.13 The Ratio Test 291 13.14 The Root Test 294 Exercises 298
Page 2 and 3: Discrete Mathematics Demystified
Page 4 and 5: Discrete Mathematics Demystified St
Page 6 and 7: To the memory of J. W. T. Youngs.
Page 8 and 9: CONTENTS Preface xiii CHAPTER 1 Log
Page 12 and 13: Contents xi Final Exam 301 Solution
Page 14 and 15: PREFACE In today’s world, analyti
Page 16 and 17: CHAPTER 1 Strictly speaking, our ap
Page 18 and 19: CHAPTER 1 Logic 3 The point of view
Page 20 and 21: CHAPTER 1 Logic 5 A B A∨ B T T T
Page 22 and 23: CHAPTER 1 Logic 7 In the next two s
Page 24 and 25: CHAPTER 1 Logic 9 the hypothesis (2
Page 26 and 27: CHAPTER 1 Logic 11 is the line in w
Page 28 and 29: CHAPTER 1 Logic 13 is the statement
Page 30 and 31: CHAPTER 1 Logic 15 and EXAMPLE 1.22
Page 32 and 33: CHAPTER 1 Logic 17 EXAMPLE 1.25 Loo
Page 34 and 35: CHAPTER 1 Logic 19 We conclude by n
Page 36 and 37: CHAPTER 1 Logic 21 6. For each of t
Page 38 and 39: CHAPTER 2 Methods of Mathematical P
Page 56 and 57: CHAPTER 3 Set Theory 3.1 Rudiments
Page 58 and 59: 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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CHAPTER 10 Cryptography 207 The fir
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CHAPTER 10 Cryptography 209 choice
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CHAPTER 10 Cryptography 211 Next le
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CHAPTER 10 Cryptography 213 we see
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CHAPTER 10 Cryptography 215 Thus we
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CHAPTER 10 Cryptography 217 Then we
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CHAPTER 10 Cryptography 219 Unfortu
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CHAPTER 10 Cryptography 221 As you
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CHAPTER 10 Cryptography 223 A typic
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CHAPTER 10 Cryptography 225 Proof:
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CHAPTER 10 Cryptography 227 Name Va
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CHAPTER 10 Cryptography 229 For sup
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CHAPTER 10 Cryptography 231 7 1 6 F
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CHAPTER 10 Cryptography 233 governm
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CHAPTER 11 Boolean Algebra 11.1 Des
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CHAPTER 11 Boolean Algebra 237 3. a
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CHAPTER 11 Boolean Algebra 239 3. [
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CHAPTER 11 Boolean Algebra 241 In s
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CHAPTER 12 Sequences 12.1 Introduct
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CHAPTER 12 Sequences 245 Remark 12.
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CHAPTER 12 Sequences 247 Figure 12.
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CHAPTER 12 Sequences 249 then we no
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CHAPTER 12 Sequences 251 3. lim j
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CHAPTER 12 Sequences 253 We conclud
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CHAPTER 12 Sequences 255 3. If t =
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CHAPTER 13 Series 13.1 Fundamental
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CHAPTER 13 Series 259 and 7 j=3 c j
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CHAPTER 13 Series 261 Insight: Do n
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CHAPTER 13 Series 263 series is equ
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CHAPTER 13 Series 265 Insight: The
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CHAPTER 13 Series 267 where the ove
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CHAPTER 13 Series 269 But we learne
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CHAPTER 13 Series 271 converges. Bu
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CHAPTER 13 Series 273 converges. Si
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CHAPTER 13 Series 275 EXAMPLE 13.22
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CHAPTER 13 Series 277 converge as N
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CHAPTER 13 Series 279 13.9.1 AN APP
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CHAPTER 13 Series 281 a 1 a 2 a 4 F
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CHAPTER 13 Series 283 13.11.1 A NEW
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CHAPTER 13 Series 285 Solution: Not
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CHAPTER 13 Series 287 This simplifi
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CHAPTER 13 Series 291 converges bec
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CHAPTER 13 Series 293 The limit of
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CHAPTER 13 Series 295 be a series.
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CHAPTER 13 Series 299 In each of Ex
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Final Exam 1. What is the contrapos
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Final Exam 303 (d) B∧ ∼B (e) A
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Final Exam 305 18. Let S be the col
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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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