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Discrete Mathematics An Open Introd
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Oscar Levin School of Mathematical
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Acknowledgements This book would no
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Preface This text aims to give an i
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How to use this book In addition to
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Contents Acknowledgements Preface H
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Contents xiii Proof by Cases . . .
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Chapter 0 Introduction and Prelimin
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0.1. What is Discrete Mathematics?
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0.2. Mathematical Statements 5 •
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0.2. Mathematical Statements 7 Impl
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0.2. Mathematical Statements 9 3. I
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0.2. Mathematical Statements 11 The
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0.2. Mathematical Statements 13 It
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0.2. Mathematical Statements 15 Inv
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0.2. Mathematical Statements 17 and
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0.2. Mathematical Statements 19 7.
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0.2. Mathematical Statements 21 12.
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0.2. Mathematical Statements 23 (b)
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0.3. Sets 25 say that the set B is
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0.3. Sets 27 We already have a lot
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0.3. Sets 29 Example 0.3.3 Let A {
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0.3. Sets 31 Operations On Sets Is
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0.3. Sets 33 You might notice that
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0.3. Sets 35 Exercises 1. Let A {1
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0.3. Sets 37 17. Let A {a, b, c, d
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0.4. Functions 39 0.4 Functions A f
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0.4. Functions 41 It would be absol
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0.4. Functions 43 We will also be i
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0.4. Functions 45 2. We are told th
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0.4. Functions 47 2. g : {1, 2, 3}
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0.4. Functions 49 Example 0.4.8 Con
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0.4. Functions 51 Exercises 1. Cons
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0.4. Functions 53 10. Suppose f : N
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0.4. Functions 55 (c) f is bijectiv
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Chapter 1 Counting One of the first
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1.1. Additive and Multiplicative Pr
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1.1. Additive and Multiplicative Pr
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1.1. Additive and Multiplicative Pr
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1.1. Additive and Multiplicative Pr
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1.1. Additive and Multiplicative Pr
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1.1. Additive and Multiplicative Pr
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1.2. Binomial Coefficients 71 Of th
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1.2. Binomial Coefficients 73 probl
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1.2. Binomial Coefficients 75 In fa
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1.2. Binomial Coefficients 77 Remem
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1.2. Binomial Coefficients 79 3. Le
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1.3. Combinations and Permutations
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1.3. Combinations and Permutations
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1.3. Combinations and Permutations
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1.3. Combinations and Permutations
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1.4. Combinatorial Proofs 89 Invest
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1.4. Combinatorial Proofs 91 Soluti
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1.4. Combinatorial Proofs 93 Exampl
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1.4. Combinatorial Proofs 95 More P
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1.4. Combinatorial Proofs 97 Since
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1.4. Combinatorial Proofs 99 to (n,
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1.4. Combinatorial Proofs 101 8. Co
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1.5. Stars and Bars 103 Investigate
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1.5. Stars and Bars 105 some stars
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1.5. Stars and Bars 107 Example 1.5
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1.5. Stars and Bars 109 (b) How man
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1.6. Advanced Counting Using PIE 11
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3.2. Proofs 215 5. N is not divisib
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3.2. Proofs 217 to prove directly,
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3.2. Proofs 219 Example 3.2.7 Prove
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3.2. Proofs 221 In fact, we can qui
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3.2. Proofs 223 Exercises 1. Consid
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3.2. Proofs 225 14. Prove that ther
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3.3. Chapter Summary 227 3.3 Chapte
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3.3. Chapter Summary 229 6. Conside
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Chapter 4 Graph Theory Investigate!
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4.1. Definitions 233 Investigate! 4
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4.1. Definitions 235 we have edges
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4.1. Definitions 237 Alternatively,
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4.1. Definitions 239 2-element subs
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4.1. Definitions 241 the vertices w
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4.1. Definitions 243 Path A path is
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4.1. Definitions 245 9. For each of
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4.2. Trees 247 Investigate! 4.2 Tre
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4.2. Trees 249 Assume T is a tree,
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4.2. Trees 251 one. Let T ′ be th
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4.2. Trees 253 are adjacent (they a
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4.2. Trees 255 Exercises 1. Which o
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4.2. Trees 257 a d b c f e 14. Give
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4.3. Planar Graphs 259 WARNING: you
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4.3. Planar Graphs 261 which says t
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4.3. Planar Graphs 263 sphere to li
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4.3. Planar Graphs 265 which will b
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4.4. Coloring 267 Investigate! 4.4
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4.4. Coloring 269 representations o
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4.4. Coloring 271 KQEJ KQEA KQEB P
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4.4. Coloring 273 We must color the
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4.4. Coloring 275 6. Prove the chro
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4.5. Euler Paths and Circuits 277 I
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4.5. Euler Paths and Circuits 279 W
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4.5. Euler Paths and Circuits 281 (
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4.6. Matching in Bipartite Graphs 2
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4.6. Matching in Bipartite Graphs 2
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4.6. Matching in Bipartite Graphs 2
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4.7. Chapter Summary 289 4.7 Chapte
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4.7. Chapter Summary 291 (b) For wh
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4.7. Chapter Summary 293 (d) Every
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Chapter 5 Additional Topics 5.1 Gen
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5.1. Generating Functions 297 shift
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5.1. Generating Functions 299 numbe
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5.1. Generating Functions 301 relat
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5.1. Generating Functions 303 and t
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5.1. Generating Functions 305 7. Us
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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5.2. Introduction to Number Theory
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Appendix A Selected Hints 0.2 Exerc
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Selected Hints 327 1.6 Exercises 1.
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Selected Hints 329 2.5.27. For the
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Selected Hints 331 3.2.16. Your fri
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Selected Hints 333 4.4.10. The chro
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Appendix B Selected Solutions 0.2.1
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Selected Solutions 337 (d) Equivale
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Selected Solutions 339 (b) We get t
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Selected Solutions 341 (d) Such an
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Selected Solutions 343 Proof. Let x
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Selected Solutions 345 (c) 2 6 −
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Selected Solutions 347 (c) To build
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Selected Solutions 349 the box. The
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Selected Solutions 351 (c) ( 16) [(
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Selected Solutions 353 (p) Neither.
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Selected Solutions 355 x + y n. So
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Selected Solutions 357 2.1.4. (a) T
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Selected Solutions 359 (b) 6n + 1,
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Selected Solutions 361 2.4.12. We h
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Selected Solutions 363 F 2k+3 −
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Selected Solutions 365 2.6.7. (a) 4
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Selected Solutions 367 3.1.16. (a)
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Selected Solutions 369 P Q R P →
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Selected Solutions 371 (b) The conv
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Selected Solutions 373 (c) Not poss
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Selected Solutions 375 number of fa
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Selected Solutions 377 4.6 Exercise
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Selected Solutions 379 (b) Now addi
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Selected Solutions 381 4.7.21. (a)
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Selected Solutions 383 5.2.6. (a) 3
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Appendix C List of Symbols Symbol D
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Index additive principle, 57 adjace
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Index 389 Goldbach conjecture, 227
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Index 391 vs combination, 84, 123,
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Index 393 set difference, 34 vertex
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Colophon This book was authored in