Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

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110 1. Counting (d) You allow yourself multiple shots, and care about the order the flavors are added?
1.6. Advanced Counting Using PIE 111 1.6 Advanced Counting Using PIE Investigate! You have 11 identical mini key-lime pies to give to 4 children. However, you don’t want any kid to get more than 3 pies. How many ways can you distribute the pies? 1. How many ways are there to distribute the pies without any restriction? 2. Let’s get rid of the ways that one or more kid gets too many pies. How many ways are there to distribute the pies if Al gets too many pies? What if Bruce gets too many? Or Cat? Or Dent? 3. What if two kids get too many pies? How many ways can this happen? Does it matter which two kids you pick to overfeed? 4. Is it possible that three kids get too many pies? If so, how many ways can this happen? 5. How should you combine all the numbers you found above to answer the original question? Suppose now you have 13 pies and 7 children. No child can have more than 2 pies. How many ways can you distribute the pies? Attempt the above activity before proceeding Stars and bars allows us to count the number of ways to distribute 10 cookies to 3 kids and natural number solutions to x + y + z 11, for example. A relatively easy modification allows us to put a lower bound restriction on these problems: perhaps each kid must get at least two cookies or x, y, z ≥ 2. This was done by first assigning each kid (or variable) 2 cookies (or units) and then distributing the rest using stars and bars. What if we wanted an upper bound restriction? For example, we might insist that no kid gets more than 4 cookies or that x, y, z ≤ 4. It turns out this is considerably harder, but still possible. The idea is to count all the distributions and then remove those that violate the condition. In other words, we must count the number of ways to distribute 11 cookies to 3 kids in which one or more of the kids gets more than 4 cookies. For any particular kid, this is not a problem; we do this using stars and bars. But
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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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Page 87 and 88: 1.2. Binomial Coefficients 71 Of th
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Page 97 and 98: 1.3. Combinations and Permutations
Page 105 and 106: 1.4. Combinatorial Proofs 89 Invest
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Page 109 and 110: 1.4. Combinatorial Proofs 93 Exampl
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Page 113 and 114: 1.4. Combinatorial Proofs 97 Since
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Page 117 and 118: 1.4. Combinatorial Proofs 101 8. Co
Page 119 and 120: 1.5. Stars and Bars 103 Investigate
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Page 151 and 152: Chapter 2 Sequences Investigate! Th
Page 153 and 154: 2.1. Describing Sequences 137 While
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Page 165 and 166: 2.2. Arithmetic and Geometric Seque
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2.3. Polynomial Fitting 161 sequenc
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2.3. Polynomial Fitting 163 a 2 7
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2.3. Polynomial Fitting 165 5. Make
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2.4. Solving Recurrence Relations 1
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2.5. Induction 177 2.5 Induction Ma
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2.5. Induction 179 2. Prove that if
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2.5. Induction 181 the next is easi
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2.5. Induction 183 But we want to k
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2.5. Induction 185 Investigate! Str
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2.5. Induction 187 a − 1 breaks;
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2.5. Induction 189 8. Zombie Euler
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23. Use induction to prove that 2.5
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2.6. Chapter Summary 193 Investigat
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2.6. Chapter Summary 195 (c) Find a
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Chapter 3 Symbolic Logic and Proofs
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3.1. Propositional Logic 199 Truth
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3.1. Propositional Logic 201 statem
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3.1. Propositional Logic 203 propos
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3.1. Propositional Logic 205 How do
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3.1. Propositional Logic 207 Beyond
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3.1. Propositional Logic 209 Exerci
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3.1. Propositional Logic 211 ∴ P
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3.2. Proofs 213 Investigate! 3.2 Pr
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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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4.7. 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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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
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Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

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