Discrete mathematics for computer scientists PDF
Preview Discrete mathematics for computer scientists
DISCRETE MATHEMATICS FOR COMPUTER SCIENTISTS This page intentionally left blank DISCRETE MATHEMATICS FOR COMPUTER SCIENTISTS Clifford Stein ColumbiaUniversity Robert L. Drysdale DartmouthCollege Kenneth Bogart Addison-Wesley Boston Columbus Indianapolis New York San Francisco UpperSaddleRiver Amsterdam CapeTown Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi MexicoCity Sao Paulo Sydney HongKong Seoul Singapore Taipei Tokyo EditorinChief: MichaelHirsch EditorialAssistant: StephanieSellinger DirectorofMarketing: MargaretWhaples MarketingCoordinator: KathrynFerranti ManagingEditor: JeffreyHolcomb ProductionProjectManager: HeatherMcNally SeniorManufacturingBuyer: CarolMelville MediaManufacturingBuyer: GinnyMichaud ArtDirector: LindaKnowles CoverDesigner: ElenaSidorova CoverArt: Veer MediaProjectManager: KatelynBoller Full-ServiceProjectManagement: BruceHobart,Laserwords Composition: Laserwords Creditsandacknowledgmentsborrowedfromothersourcesandreproduced,withpermission,inthistextbookappearonappropriate pagewithintext. Theprogramsandapplicationspresentedinthisbookhavebeenincludedfortheirinstructionalvalue.Theyhavebeentestedwith care,butarenotguaranteedforanyparticularpurpose.Thepublisherdoesnotofferanywarrantiesorrepresentations,nordoesit acceptanyliabilitieswithrespecttotheprogramsorapplications. Copyright©2011.PearsonEducation,Inc.,publishingasAddison-Wesley,501BoylstonStreet,Suite900,Boston,Massachusetts 02116.Allrightsreserved.ManufacturedintheUnitedStatesofAmerica.ThispublicationisprotectedbyCopyright,and permissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmission inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Toobtainpermission(s)tousematerial fromthiswork,pleasesubmitawrittenrequesttoPearsonEducation,Inc.,PermissionsDepartment,501BoylstonStreet,Suite 900,Boston,Massachusetts02116. Manyofthedesignationsusedbymanufacturersandsellerstodistinguishtheirproductsareclaimedastrademarks.Wherethose designationsappearinthisbook,andthepublisherwasawareofatrademarkclaim,thedesignationshavebeenprintedininitial capsorallcaps. LibraryofCongressCataloging-in-PublicationData 10 9 8 7 6 5 4 3 2 1 ISBN-13:978-0-13-212271-9 ISBN-10:0-13-212271-5 This book is dedicated to our friend and co-author, Ken Bogart, whose untimely death on March 30, 2005 prevented him from seeing the original book in published form. Ken was the driving force behind the creation of the book. We miss him and we wish that we had been able to collaborate with him on this version. This page intentionally left blank Brief Contents List of Theorems, Lemmas, and Corollaries xix Preface xxi CHAPTER 1 Counting 1 CHAPTER 2 Cryptography and Number Theory 59 CHAPTER 3 Reflections on Logic and Proof 117 CHAPTER 4 Induction, Recursion, and Recurrences 161 CHAPTER 5 Probability 249 CHAPTER 6 Graphs 359 APPENDIX A Derivation of the More General Master Theorem 449 APPENDIX B Answers and Hints to Selected Problems 461 Bibliography 477 Index 479 vii This page intentionally left blank Contents List of Theorems, Lemmas, and Corollaries xix Preface xxi CHAPTER 1 Counting 1 1.1 Basic Counting 1 The Sum Principle 1 Abstraction 3 Summing Consecutive Integers 3 The Product Principle 4 Two-Element Subsets 6 Important Concepts, Formulas, and Theorems 7 Problems 8 1.2 Counting Lists, Permutations, and Subsets 10 Using the Sum and Product Principles 10 Lists and Functions 12 The Bijection Principle 14 k-Element Permutations of a Set 15 Counting Subsets of a Set 16 Important Concepts, Formulas, and Theorems 18 Problems 20 1.3 Binomial Coefficients 22 Pascal’s Triangle 22 A Proof Using the Sum Principle 24 The Binomial Theorem 26 Labeling and Trinomial Coefficients 28 Important Concepts, Formulas, and Theorems 29 Problems 30 ix
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