Discrete Mathematics Demystified

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CHAPTER 7 Matrices 139 and Calculate both A · B and B · A. Solution: Now Also ⎛ 3 2 ⎞ 1 B = ⎝ 4 5 6⎠ −8 0 −2 ⎛ −13 12 ⎞ 7 A · B = ⎝ 6 37 28⎠ 58 −4 14 ⎛ 13 16 ⎞ 9 B · A = ⎝ 22 33 −16⎠ −4 −16 −8 One immediate lesson here is that A · B = B · A. Multiplication of matrices is not commutative. A second lesson is that the only time we can calculate both A · B and B · A is when both matrics are square and both are of the same size. 7.3 Gaussian Elimination In high school algebra everyone learns how to solve a system of two linear equations in two unknowns: ax + by = α cx + dy = β You simply multiply the first equation by a constant so that its x-coefficient matches the x-coefficient of the second equation. Then subtraction eliminates the x-variable and one can solve directly for y. Reverse substitution then yields the value of the x-variable, and the system is solved.
140 Discrete Mathematics Demystified Figure 7.1 Typical (empty) intersection of three lines in the plane. Matters are more complicated for systems of three equations in three unknowns, or more generally k equations in k unknowns. It becomes difficult to keep track of the information, and the attendant calculations can be daunting. In this section we introduce the method of Gaussian elimination, which gives a straightforward and virtually mechanical method for solving systems of linear equations. In fact it is straightforward to implement the method of Gaussian elimination on a computer; for a system of k equations in k unknowns it takes about k 3 calculations to find the solution. So this is a robust and efficient technique. We concentrate our efforts on systems of k equations in k unknowns because such a system will generically have a unique solution (that is, a single value for each of the variables that solves the system). When there are fewer unknowns than equations then the system will typically have no solutions. 1 When there are more unknowns than equations then the system will typically have an entire space of solutions. For example, the solution of the system x + z = 4 z = 3 1 Think of intersecting lines in the plane. Two lines in the plane will usually have a point of intersection—as long as they are not parallel. Three lines in the plane will usually not have a mutual point of intersection—see Fig. 7.1.
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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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Page 150 and 151: CHAPTER 7 Matrices 7.1 What Is a Ma
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Page 156 and 157: CHAPTER 7 Matrices 141 are all poin
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Page 160 and 161: CHAPTER 7 Matrices 145 Translating
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Page 170 and 171: CHAPTER 7 Matrices 155 Thus we disc
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Page 176 and 177: CHAPTER 7 Matrices 161 f (0, 1, 1)
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 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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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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