Discrete Mathematics Demystified

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CHAPTER 7 Matrices 161 f (0, 1, 1) = 3 f (1, 0, 1) = 10 f (1, 1, 0) =−1 f (0, 0, 0) = 0 f (1, 1, 1) = 6 We see that the greatest value of f is 10, and it is assumed at the point (1, 0, 1). The least value of f is −4, and it is assumed at the point (0, 1, 0). Exercises In Exercises 1 and 2, say what the dimensions of the matrix are. ⎛ 3 ⎜ 4 1. ⎜ ⎝ 2 2 −9 1 ⎞ 7 −2 ⎟ 6⎠ 5 ⎛ 3 0 5 0 −4 2 ⎞ 0 2. ⎝ 1 1 9 4 −2 ⎠ 6 4 3 2 −1 In Exercises 3, 4, 5, and 6, calculate the indicated matrix operation. 3 3. 2 −2 9 5 −6 + 6 7 2 −4 8 9 6 4. 3 5 5. 1 −5 4 −2 4 ⎛ ⎞ 6 −5 4 · ⎝ 3 4⎠ 4 −2 2 3 6 8 −1 − 9 0 2 0 6. 2 −3 4 −7 · −5 1 −9 6
162 Discrete Mathematics Demystified 7. Calculate the inverse of the matrix ⎛ 1 0 ⎞ 1 ⎝ 0 1 1⎠ 2 1 0 8. Without actually calculating the inverse, say whether this matrix has an inverse: ⎛ 3 2 ⎞ 1 ⎝ −1 −3 −2 ⎠ 0 1 6 9. An unfair coin has twice the likelihood of landing heads as tails. Describe the coin flips as a Markov process.
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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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Page 150 and 151: CHAPTER 7 Matrices 7.1 What Is a Ma
Page 152 and 153: CHAPTER 7 Matrices 137 EXAMPLE 7.2
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Page 160 and 161: CHAPTER 7 Matrices 145 Translating
Page 162 and 163: CHAPTER 7 Matrices 147 If we guess
Page 164 and 165: CHAPTER 7 Matrices 149 Solution: We
Page 166 and 167: CHAPTER 7 Matrices 151 Finally we m
Page 168 and 169: CHAPTER 7 Matrices 153 7.5 Markov C
Page 170 and 171: CHAPTER 7 Matrices 155 Thus we disc
Page 172 and 173: CHAPTER 7 Matrices 157 Figure 7.2 A
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Page 178 and 179: CHAPTER 8 Graph Theory 8.1 Introduc
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Page 182 and 183: CHAPTER 8 Graph Theory 167 Figure 8
Page 184 and 185: CHAPTER 8 Graph Theory 169 8.3 Appl
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Page 192 and 193: CHAPTER 8 Graph Theory 177 number o
Page 194 and 195: CHAPTER 8 Graph Theory 179 $65 D A
Page 196 and 197: CHAPTER 8 Graph Theory 181 Exercise
Page 198 and 199: CHAPTER 9 Number Theory 9.1 Divisib
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Page 214 and 215: CHAPTER 10 Cryptography 10.1 Backgr
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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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