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# Discrete Mathematics and its Applications (Rosen)7th ed McGraw Hill 2012-- csc 245

## Discrete Mathematics and its Applications (Rosen)7th ed McGraw Hill 2012-- csc

Kenneth H. Rosen Discrete Mathematics and Its Applications SEVENTH EDITION

• Page 2 and 3: Discrete Mathematics and Its Applic
• Page 4 and 5: Contents About the Author vi Prefac
• Page 6 and 7: Contents v 10 Graphs ..............
• Page 8 and 9: Preface In writing this book, I was
• Page 10 and 11: Preface ix Separate chapters now p
• Page 12 and 13: Preface xi FLEXIBILITY This text ha
• Page 14 and 15: Preface xiii instructor. A two-term
• Page 16 and 17: Preface xv Darrell Minor Columbus S
• Page 18 and 19: The Companion Website xvii covering
• Page 20 and 21: To the Student xix percentage of th
• Page 22 and 23: C H A P T E R 1 The Foundations: Lo
• Page 24 and 25: 1.1 Propositional Logic 3 conventio
• Page 26 and 27: 1.1 Propositional Logic 5 The use o
• Page 28 and 29: 1.1 Propositional Logic 7 You might
• Page 30 and 31: 1.1 Propositional Logic 9 EXAMPLE 9
• Page 32 and 33: 1.1 Propositional Logic 11 Preceden
• Page 34 and 35: 1.1 Propositional Logic 13 5. What
• Page 36 and 37: 1.1 Propositional Logic 15 24. Writ
• Page 38 and 39: 1.2 Applications of Propositional L
• Page 40 and 41: 1.2 Applications of Propositional L
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• Page 46 and 47: 1.3 Propositional Equivalences 25 1
• Page 48 and 49: 1.3 Propositional Equivalences 27 T
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1.3 Propositional Equivalences 31 s

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1.3 Propositional Equivalences 33 T

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1.3 Propositional Equivalences 35 c

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1.4 Predicates and Quantifiers 37 N

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1.4 Predicates and Quantifiers 39 E

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1.4 Predicates and Quantifiers 41 T

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1.4 Predicates and Quantifiers 43 T

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1.4 Predicates and Quantifiers 45 T

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1.4 Predicates and Quantifiers 47 T

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1.4 Predicates and Quantifiers 49 I

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1.4 Predicates and Quantifiers 51 S

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1.4 Predicates and Quantifiers 53 E

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1.4 Predicates and Quantifiers 55 3

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1.5 Nested Quantifiers 57 62. Let P

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1.5 Nested Quantifiers 59 and both

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1.5 Nested Quantifiers 61 Solution:

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1.5 Nested Quantifiers 63 [Note tha

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1.5 Nested Quantifiers 65 given at

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1.5 Nested Quantifiers 67 c) The su

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1.6 Rules of Inference 69 1.6 Rules

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1.6 Rules of Inference 71 Rules of

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1.6 Rules of Inference 73 EXAMPLE 5

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1.6 Rules of Inference 75 EXAMPLE 9

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1.6 Rules of Inference 77 EXAMPLE 1

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1.6 Rules of Inference 79 10. For e

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1.7 Introduction to Proofs 81 theor

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1.7 Introduction to Proofs 83 DEFIN

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1.7 Introduction to Proofs 85 EXAMP

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1.7 Introduction to Proofs 87 We ha

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1.7 Introduction to Proofs 89 Mista

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1.7 Introduction to Proofs 91 Exerc

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1.8 Proof Methods and Strategy 93 E

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1.8 Proof Methods and Strategy 95 C

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1.8 Proof Methods and Strategy 97 B

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1.8 Proof Methods and Strategy 99 (

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1.8 Proof Methods and Strategy 101

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1.8 Proof Methods and Strategy 103

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1.8 Proof Methods and Strategy 105

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1.8 Proof Methods and Strategy 107

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Key Terms and Results 109 33. Adapt

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Supplementary Exercises 111 RESULTS

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Supplementary Exercises 113 23. Fin

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C H A P T E R 2 Basic Structures: S

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2.1 Sets 117 (Note that some people

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2.1 Sets 119 Subsets It is common t

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2.1 Sets 121 Showing Two Sets are E

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2.1 Sets 123 Many of the discrete s

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2.1 Sets 125 Truth Sets and Quantif

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2.2 Set Operations 127 2.2 Set Oper

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2.2 Set Operations 129 U U A B A A

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2.2 Set Operations 131 EXAMPLE 11 U

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2.2 Set Operations 133 EXAMPLE 15 L

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2.2 Set Operations 135 Solution: Th

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2.2 Set Operations 137 30. Can you

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2.3 Functions 139 Adams Chou Goodfr

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2.3 Functions 141 DEFINITION 3 Let

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2.3 Functions 143 a b c d FIGURE 4

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2.3 Functions 145 Suppose that f :

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2.3 Functions 147 ( f g)(a) g(a) f(

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2.3 Functions 149 DEFINITION 12 The

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2.3 Functions 151 We first consider

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2.3 Functions 153 7. Find the domai

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2.3 Functions 155 59. How many byte

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2.4 Sequences and Summations 157 DE

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2.4 Sequences and Summations 159 EX

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2.4 Sequences and Summations 161 Wh

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2.4 Sequences and Summations 163 fr

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2.4 Sequences and Summations 165 To

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2.4 Sequences and Summations 167 SO

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2.4 Sequences and Summations 169 a)

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2.5 Cardinality of Sets 171 1 2 3 4

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2.5 Cardinality of Sets 173 Terms n

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2.5 Cardinality of Sets 175 Because

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2.6 Matrices 177 18. Show that if A

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2.6 Matrices 179 We now discuss mat

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2.6 Matrices 181 DEFINITION 6 EXAMP

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2.6 Matrices 183 DEFINITION 10 Let

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Key Terms and Results 185 24. a) Sh

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Supplementary Exercises 187 7. Expl

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Computer Projects 189 Computer Proj

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192 3 / Algorithms The term algorit

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194 3 / Algorithms of the terms of

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196 3 / Algorithms Algorithm 3 proc

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198 3 / Algorithms is not less than

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200 3 / Algorithms Proof: We will u

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202 3 / Algorithms Input Program P

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204 3 / Algorithms 51. When a list

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206 3 / Algorithms THE HISTORY OF B

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208 3 / Algorithms Cg(x) f (x) g(x)

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210 3 / Algorithms where C =|a n |+

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212 3 / Algorithms USEFUL BIG-O EST

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214 3 / Algorithms EXAMPLE 8 Give a

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216 3 / Algorithms One useful fact

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218 3 / Algorithms 59. (Requires ca

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220 3 / Algorithms EXAMPLE 2 Descri

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222 3 / Algorithms using a summatio

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224 3 / Algorithms multiplications

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226 3 / Algorithms TABLE 1 Commonly

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230 3 / Algorithms a) Show that thi

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232 3 / Algorithms Key Terms and Re

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234 3 / Algorithms 21. Find all pai

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236 3 / Algorithms 9. Given an orde

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238 4 / Number Theory and Cryptogra

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240 4 / Number Theory and Cryptogra

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242 4 / Number Theory and Cryptogra

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244 4 / Number Theory and Cryptogra

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246 4 / Number Theory and Cryptogra

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248 4 / Number Theory and Cryptogra

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250 4 / Number Theory and Cryptogra

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256 4 / Number Theory and Cryptogra

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260 4 / Number Theory and Cryptogra

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266 4 / Number Theory and Cryptogra

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270 4 / Number Theory and Cryptogra

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272 4 / Number Theory and Cryptogra

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274 4 / Number Theory and Cryptogra

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276 4 / Number Theory and Cryptogra

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280 4 / Number Theory and Cryptogra

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284 4 / Number Theory and Cryptogra

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286 4 / Number Theory and Cryptogra

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288 4 / Number Theory and Cryptogra

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290 4 / Number Theory and Cryptogra

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292 4 / Number Theory and Cryptogra

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294 4 / Number Theory and Cryptogra

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296 4 / Number Theory and Cryptogra

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298 4 / Number Theory and Cryptogra

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300 4 / Number Theory and Cryptogra

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302 4 / Number Theory and Cryptogra

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304 4 / Number Theory and Cryptogra

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306 4 / Number Theory and Cryptogra

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308 4 / Number Theory and Cryptogra

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310 4 / Number Theory and Cryptogra

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312 5 / Induction and Recursion We

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314 5 / Induction and Recursion FIG

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316 5 / Induction and Recursion EXA

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318 5 / Induction and Recursion EXA

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320 5 / Induction and Recursion tha

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322 5 / Induction and Recursion ste

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324 5 / Induction and Recursion IND

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326 5 / Induction and Recursion to

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328 5 / Induction and Recursion To

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330 5 / Induction and Recursion 9.

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332 5 / Induction and Recursion of

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334 5 / Induction and Recursion two

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336 5 / Induction and Recursion We

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338 5 / Induction and Recursion IND

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340 5 / Induction and Recursion a b

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342 5 / Induction and Recursion par

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344 5 / Induction and Recursion ∗

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346 5 / Induction and Recursion EXA

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348 5 / Induction and Recursion Pro

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350 5 / Induction and Recursion The

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352 5 / Induction and Recursion Bas

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354 5 / Induction and Recursion In

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356 5 / Induction and Recursion If

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358 5 / Induction and Recursion 7.

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360 5 / Induction and Recursion ∗

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362 5 / Induction and Recursion EXA

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364 5 / Induction and Recursion pro

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366 5 / Induction and Recursion f 4

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368 5 / Induction and Recursion alg

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370 5 / Induction and Recursion by

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372 5 / Induction and Recursion 5.5

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374 5 / Induction and Recursion EXA

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376 5 / Induction and Recursion Let

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378 5 / Induction and Recursion str

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380 5 / Induction and Recursion 30.

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382 5 / Induction and Recursion The

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C H A P T E R 6 Counting 6.1 The Ba

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6.1 The Basics of Counting 387 EXAM

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6.1 The Basics of Counting 389 Soon

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6.1 The Basics of Counting 391 More

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6.1 The Basics of Counting 393 THE

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6.1 The Basics of Counting 395 Winn

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6.1 The Basics of Counting 397 25.

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6.2 The Pigeonhole Principle 399 71

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6.2 The Pigeonhole Principle 401 Th

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6.2 The Pigeonhole Principle 403 wi

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6.2 The Pigeonhole Principle 405 pe

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6.3 Permutations and Combinations 4

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6.3 Permutations and Combinations 4

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6.3 Permutations and Combinations 4

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6.3 Permutations and Combinations 4

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6.4 Binomial Coefficients and Ident

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6.4 Binomial Coefficients and Ident

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6.4 Binomial Coefficients and Ident

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6.4 Binomial Coefficients and Ident

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6.5 Generalized Permutations and Co

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6.5 Generalized Permutations and Co

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6.5 Generalized Permutations and Co

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6.5 Generalized Permutations and Co

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6.5 Generalized Permutations and Co

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6.5 Generalized Permutations and Co

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6.6 Generating Permutations and Com

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6.6 Generating Permutations and Com

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Review Questions 439 16. Find the p

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Supplementary Exercises 441 3. A te

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Supplementary Exercises 443 45. How

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C H A P T E R 7 Discrete Probabilit

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7.1 An Introduction to Discrete Pro

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7.1 An Introduction to Discrete Pro

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7.1 An Introduction to Discrete Pro

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7.2 Probability Theory 453 similar

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7.2 Probability Theory 455 Solution

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7.2 Probability Theory 457 EXAMPLE

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7.2 Probability Theory 459 EXAMPLE

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7.2 Probability Theory 461 Solution

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7.2 Probability Theory 463 It follo

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7.2 Probability Theory 465 can use

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7.2 Probability Theory 467 10. What

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7.3 Bayes’ Theorem 469 this disea

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7.3 Bayes’ Theorem 471 EXAMPLE 2

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7.3 Bayes’ Theorem 473 We will de

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7.3 Bayes’ Theorem 475 event that

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7.4 Expected Value and Variance 477

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7.4 Expected Value and Variance 479

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7.4 Expected Value and Variance 481

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7.4 Expected Value and Variance 483

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7.4 Expected Value and Variance 485

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7.4 Expected Value and Variance 487

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7.4 Expected Value and Variance 489

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7.4 Expected Value and Variance 491

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7.4 Expected Value and Variance 493

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7.4 Expected Value and Variance 495

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Supplementary Exercises 497 b) What

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Supplementary Exercises 499 ∗36.

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C H A P T E R 8 Advanced Counting T

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8.1 Applications of Recurrence Rela

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8.1 Applications of Recurrence Rela

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8.1 Applications of Recurrence Rela

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8.1 Applications of Recurrence Rela

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8.1 Applications of Recurrence Rela

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8.1 Applications of Recurrence Rela

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8.2 Solving Linear Recurrence Relat

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8.2 Solving Linear Recurrence Relat

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8.2 Solving Linear Recurrence Relat

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8.2 Solving Linear Recurrence Relat

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8.2 Solving Linear Recurrence Relat

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8.2 Solving Linear Recurrence Relat

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8.3 Divide-and-Conquer Algorithms a

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8.3 Divide-and-Conquer Algorithms a

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8.3 Divide-and-Conquer Algorithms a

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8.3 Divide-and-Conquer Algorithms a

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8.3 Divide-and-Conquer Algorithms a

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8.4 Generating Functions 537 8.4 Ge

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8.4 Generating Functions 539 Remark

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8.4 Generating Functions 541 Using

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8.4 Generating Functions 543 This f

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8.4 Generating Functions 545 EXAMPL

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8.4 Generating Functions 547 Solvin

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8.4 Generating Functions 549 Exerci

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8.4 Generating Functions 551 35. Us

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8.5 Inclusion-Exclusion 553 Section

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8.5 Inclusion-Exclusion 555 1 1 1 1

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8.5 Inclusion-Exclusion 557 Therefo

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8.6 Applications of Inclusion-Exclu

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8.6 Applications of Inclusion-Exclu

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8.6 Applications of Inclusion-Exclu

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Key Terms and Results 565 11. In ho

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Supplementary Exercises 567 b) How

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Supplementary Exercises 569 31. (Re

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Computations and Explorations 571 9

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574 9 / Relations (Jason Goodfriend

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576 9 / Relations EXAMPLE 6 How man

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578 9 / Relations EXAMPLE 12 Is the

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580 9 / Relations DEFINITION 6 Let

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582 9 / Relations A relation R is c

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584 9 / Relations school are sophom

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586 9 / Relations EXAMPLE 6 Is the

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588 9 / Relations TABLE 5 Teaching_

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590 9 / Relations 9. The 5-tuples i

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592 9 / Relations Solution: Because

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594 9 / Relations The matrix repres

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596 9 / Relations Because loops are

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598 9 / Relations of R with respect

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600 9 / Relations The term path als

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602 9 / Relations x i+2 x i+1 x j-2

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604 9 / Relations a EXAMPLE 8 b are

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606 9 / Relations LEMMA 2 Let W k =

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608 9 / Relations uppercase or lowe

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610 9 / Relations EXAMPLE 6 In Exam

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612 9 / Relations Equivalence Class

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614 9 / Relations Solution: The sub

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616 9 / Relations In Exercises 21-2

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618 9 / Relations ∗58. Each bead

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620 9 / Relations EXAMPLE 6 The pos

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622 9 / Relations EXAMPLE 10 EXAMPL

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624 9 / Relations 8 12 8 12 8 12 4

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626 9 / Relations EXAMPLE 19 Find t

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628 9 / Relations cannot be started

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630 9 / Relations Exercises 1. Whic

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632 9 / Relations 40. a) Show that

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634 9 / Relations lexicographic ord

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636 9 / Relations 15. a) Give an ex

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638 9 / Relations ∗48. Show that

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C H A P T E R 10 Graphs 10.1 Graphs

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10.1 Graphs and Graph Models 643 Ch

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10.1 Graphs and Graph Models 645 Ed

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10.1 Graphs and Graph Models 647 fi

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10.1 Graphs and Graph Models 649 Te

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.2 Graph Terminology and Special

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10.3 Representing Graphs and Graph

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10.3 Representing Graphs and Graph

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10.3 Representing Graphs and Graph

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10.3 Representing Graphs and Graph

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10.3 Representing Graphs and Graph

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10.4 Connectivity 679 A formal defi

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10.4 Connectivity 681 TABLE 1 The N

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10.4 Connectivity 683 Sometimes the

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10.4 Connectivity 685 The graph G 2

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10.4 Connectivity 687 GSCC followin

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10.4 Connectivity 689 there are exa

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10.4 Connectivity 691 22. Use paths

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10.5 Euler and Hamilton Paths 693

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10.5 Euler and Hamilton Paths 695 a

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10.5 Euler and Hamilton Paths 697 a

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10.5 Euler and Hamilton Paths 699 (

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10.5 Euler and Hamilton Paths 701 i

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10.5 Euler and Hamilton Paths 703 T

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10.5 Euler and Hamilton Paths 705 2

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10.6 Shortest-Path Problems 707 A k

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10.6 Shortest-Path Problems 709 DIS

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10.6 Shortest-Path Problems 711 To

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10.6 Shortest-Path Problems 713 b

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10.6 Shortest-Path Problems 715 Rou

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10.6 Shortest-Path Problems 717 17.

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10.7 Planar Graphs 719 FIGURE 2 Gra

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10.7 Planar Graphs 721 R 1 a 1 b 1

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10.7 Planar Graphs 723 G 1 a b G 2

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10.7 Planar Graphs 725 a e j f g b

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10.8 Graph Coloring 727 A B C D E F

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10.8 Graph Coloring 729 b e b e a d

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10.8 Graph Coloring 731 Next, let n

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10.8 Graph Coloring 733 3. 11. e h

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Key Terms and Results 735 ∗37. Le

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Review Questions 737 chromatic numb

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Supplementary Exercises 739 10. Let

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Supplementary Exercises 741 38. Bec

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Writing Projects 743 18. Given the

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C H A P T E R 11 Trees 11.1 Introdu

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11.1 Introduction to Trees 747 Proo

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11.1 Introduction to Trees 749 Solu

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11.1 Introduction to Trees 751 Pres

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11.1 Introduction to Trees 753 THEO

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11.1 Introduction to Trees 755 Then

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11.2 Applications of Trees 757 The

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11.2 Applications of Trees 759 ALGO

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11.2 Applications of Trees 761 1 2

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11.2 Applications of Trees 763 Each

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11.2 Applications of Trees 765 0.08

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11.2 Applications of Trees 767 (a)

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11.2 Applications of Trees 769 each

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11.2 Applications of Trees 771 27.

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11.3 Tree Traversal 773 0 1 2 3 4 5

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11.3 Tree Traversal 775 r Step 2: V

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11.3 Tree Traversal 777 r Step n +1

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11.3 Tree Traversal 779 ALGORITHM 2

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11.3 Tree Traversal 781 + - * 2 3 5

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11.3 Tree Traversal 783 Exercises I

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11.4 Spanning Trees 785 Herman Etna

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11.4 Spanning Trees 787 Source IP n

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11.4 Spanning Trees 789 explore fro

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11.4 Spanning Trees 791 ALGORITHM 2

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11.4 Spanning Trees 793 X X X X X X

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11.4 Spanning Trees 795 links at th

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11.5 Minimum Spanning Trees 797 41.

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11.5 Minimum Spanning Trees 799 San

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11.5 Minimum Spanning Trees 801 e 3

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Key Terms and Results 803 15. Find

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Supplementary Exercises 805 root, t

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Supplementary Exercises 807 33. d a

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Writing Projects 809 5. Define quad

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812 12 / Boolean Algebra The comple

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814 12 / Boolean Algebra TABLE 3 Th

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816 12 / Boolean Algebra complement

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818 12 / Boolean Algebra Exercises

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820 12 / Boolean Algebra DEFINITION

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822 12 / Boolean Algebra Exercises

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824 12 / Boolean Algebra x y xy xy

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826 12 / Boolean Algebra x y z x y

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828 12 / Boolean Algebra 5. x y z x

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830 12 / Boolean Algebra second pro

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832 12 / Boolean Algebra yz yz yz y

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834 12 / Boolean Algebra yz yz yz y

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836 12 / Boolean Algebra EXAMPLE 7

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838 12 / Boolean Algebra TABLE 2 Mi

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840 12 / Boolean Algebra 4. Determi

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842 12 / Boolean Algebra c) x y z x

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844 12 / Boolean Algebra An identit

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846 12 / Boolean Algebra ∗7. Give

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848 13 / Modeling Computation to an

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850 13 / Modeling Computation EXAMP

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852 13 / Modeling Computation lengt

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854 13 / Modeling Computation that

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856 13 / Modeling Computation adjec

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858 13 / Modeling Computation Sever

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860 13 / Modeling Computation R, n

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862 13 / Modeling Computation Start

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864 13 / Modeling Computation c) f

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866 13 / Modeling Computation DEFIN

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868 13 / Modeling Computation We ca

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870 13 / Modeling Computation (b) O

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872 13 / Modeling Computation a 1,

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874 13 / Modeling Computation Start

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876 13 / Modeling Computation 15. G

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878 13 / Modeling Computation or bo

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880 13 / Modeling Computation Solut

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882 13 / Modeling Computation (a) T

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884 13 / Modeling Computation Start

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886 13 / Modeling Computation Alan

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888 13 / Modeling Computation 20. S

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890 13 / Modeling Computation (a) s

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892 13 / Modeling Computation the s

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894 13 / Modeling Computation Many

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896 13 / Modeling Computation COMPU

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898 13 / Modeling Computation 3. Wh

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900 13 / Modeling Computation decis

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902 13 / Modeling Computation and t

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A P P E N D I X 1 Axioms for the Re

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Appendix 1 / Axioms for the Real Nu

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Appendix 1 / Axioms for the Real Nu

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A P P E N D I X 2 Exponential and L

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Appendix 2 / Exponential and Logari

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A-12 Appendix 3 / Pseudocode For ex

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A-14 Appendix 3 / Pseudocode Loop C

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A-16 Appendix 3 / Pseudocode Exerci

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B-6 Suggested Reading [Ha93] John P

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B-8 Suggested Reading [SePi89] J. S

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Photo Credits CHAPTER 1 Page 2: ©

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Index of Biographies Ada, Augusta (

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Index I-3 Antisymmetric relation, 5

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Index I-5 Climbing rock, 163 Clique

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Index I-7 Diagrams Hasse, 622-626,

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Index I-9 Folder empty, 118 Forbidd

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Index I-11 Identification number si

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Index I-13 Linear array, 662 Linear

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Index I-15 Notation big-O, 205, 232

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Index I-17 Millennium Prize, 227 Ne

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Index I-19 S k -tree, 806 Same pari

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Index I-21 Summation index of, 163

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Index I-23 Vertex (vertices)—Cont

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