Discrete Mathematics for Computer Science

280 Pages·2022·15.385 MB·English

by  Alexander Golovnev, Alexander S. Kulikov, Vladimir V. Podolskii, Alexander Shen

DISCRETE
MATHEMATICS
for Computer Science
Alexander Golovnev, Alexander S. Kulikov, Vladimir V. Podolskii, Alexander Shen
Welcome!
Thankyoufordownloadingthisbook!ItsupplementstheIntroductiontoDiscreteMathematicsfor
ComputerScience specializationatCourseraandcontainsmanyinteractivepuzzles,autograded
quizzes,andcodesnippets.Theyareintendedtohelpyoutodiscoverimportantideasindiscrete
mathematicsonyourown,andtoshowyoucorrespondingapplicationsoftheseideasincomputer
science.
This book contains material corresponding to the first two courses in the associated spe-
cializationatCoursera:MathematicalThinkinginComputerScience andCombinatoricsand
Probability . Futureeditionswillcovertheadditionalthreecourses: GraphTheory, Number
TheoryandCryptography,andDeliveryProblem.
Thereare300+problemsand150+codesnippetsinthebook.Mostoftheproblemscomewith
solutionsand130+ofthemaregradedautomatically(allowingyoutogetinstantfeedback).We’re
constantlyworkingonextendingandimprovingthisbook.Pleaseaskquestions,reporttypos,and
suggestimprovementsthroughthisform.Checkhttps://leanpub.com/discrete-math forupdates
ofthebook.
Contents
0 About the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.1 ActiveLearning 7
0.2 Problem-basedLearning 7
0.3 PythonProgrammingLanguage 8
0.4 Acknowledgments 9
I Mathematical Thinking in Computer Science
1 Proofs: Convincing Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 WarmUp 13
1.2 ExistenceProofs 20
2 Finding an Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 HowtoFindanExample 31
2.2 Optimality 39
2.3 ComputerSearch 49
3 Recursion and Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Recursion 57
3.2 Induction 75
4 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1 ExamplesandCounterexamples 93
4.2 Logic 97
4.3 ReductioadAbsurdum 103
5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1 DoubleCounting 109
5.2 SearchingforInvariants 111
5.3 Termination 112
5.4 EvenandOddNumbers 115
6 Project: 15-Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 ThePuzzle 121
6.2 PermutationsandTranspositions 122
6.3 Why15-puzzleHasNoSolution 133
6.4 When15-puzzleHasaSolution 135
6.5 Implementation 142
7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1 CuttingaFigure 149
7.2 UsingSAT-solvers 149
7.3 UsingILP-solvers 153
7.4 VisualizingFootballFans 156
II Combinatorics and Probability
8 Basic Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1 StartingtoCount 161
8.2 RecursiveCounting 167
8.3 TuplesandPermutations 173
9 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.1 NumberofGamesinaTournament 181
9.2 Combinations 184
9.3 BinomialTheorem 190
9.4 PracticeCounting 193
10 Advanced Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.1 Review 197
10.2 CombinationswithRepetitions 198
10.3 PracticeCounting 202
11 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.1 WhatisProbability? 211
11.2 Probability:Do’sandDon’ts 227
11.3 ConditionalProbability 241
11.4 MontyHallParadox 253
12 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
12.1 RandomVariablesandTheirExpectations 259
12.2 LinearityofExpectation 268
12.3 ExpectationisNotAll 271
12.4 Markov’sInequality 273
13 Dice Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
13.1 DiceGameProblem 275
13.2 OptimalStrategy 278

0. About the Book
0.1 Active Learning
Thisbookcoversideasandconceptsindiscretemathematicswhichareneededinvariousbranches
ofcomputerscience.Tomakethelearningprocessmoreefficientandenjoyable,weusethefol-
lowingactivelearningcomponentsimplementedthroughourIntroductiontoDiscreteMathematics
forComputerSciencespecialization atCoursera.
Interactivepuzzles provide you with a fun way to “invent” the key ideas on your own. The
puzzlesaremobile-friendly,soyoucanplaywiththemanywhere.Thegoalofeverypuzzle
is to give you a clean and easy way to state problems where nothing distracts you from
inventingamethodforsolvingit. Inturn,thecorrespondingmethodusuallyhasawide
rangeofapplicationstovariousproblemsincomputerscience.
Autogradedquizzes allowyoutoimmediatelycheckyourunderstandingafterlearninganew
conceptoridea.
Codesnippets arehelpfulintwoways:1)theyshowyouhowideasfromdiscretemathematics
areusedinprogramming,and2)theyserveasinteractiveexamplesandchallenges:tweak
thegivenpieceofcode,runit,andseewhathappens.
Programmingchallenges willhelpyoutosolidifyyourunderstanding.AsDonaldKnuthsaid,
“IfindthatIdon’tunderstandthingsunlessItrytoprogramthem.”
0.2 Problem-based Learning
Throughoutthebook(andtheassociatedspecializationatCoursera)wefollowa“trythisbefore
weexplaineverything”approach:wealwaysaskyoutosolveaproblemfirst,andthenweexplain
howtosolveitandintroduceimportantideasneededtosolveit.Webelieve,thiswayyouwillget
adeeperunderstandingandalsodevelopabetterappreciationforthebeautyoftheunderlying
ideas(nottomentiontheself-confidencethatyougetifyouinventtheseideasonyourown!).
Don’tbediscouragedifyoucan’tsolvealltheproblems. Justhavingattemptedthemisoften
enoughtoengageyourmind,andmakeyoumorecuriousaboutthesolution.
Weusethefollowingtwobasictypesofquestionsinthebook.
Stopandthink questionsinviteyoutoslowdownandcontemplatethecurrentmaterialbefore
continuingtothenexttopic.Wealwaysprovideananswertothecorrespondingquestion
rightafterit. Westronglyencourageyou(asthenamesuggests)tostopanddoyourbest
toanswerthequestion.
8 Chapter0.AbouttheBook
Problems usuallyrequiremoreefforttosolve.Weusesomeofthemtowarmyouupandtodevelop
yourcuriosity.Suchproblemsarefollowedbydetailedsolutions.Someotherproblemsare
leftforyouasexercises.
ManyquestionsinthebookaregradedautomaticallythroughCoursera.Theyaremarkedwith:
Tryit:Coursera ,external .
Boththeselinksareclickable:thefirstoneopensthecorrespondingautogradedpuzzleatCoursera
(thisrequiresanactivesubscriptiontothespecialization),thesecondoneopensthecorresponding
interactivepuzzle(andrequiresnosubscription).Atthesametime,thebookisself-contained:if
youareunabletowatchthevideosandaccesstheinteractivepuzzlesatCoursera,justreadthe
bookandsolvetheproblemsonapieceofpaper.
0.3 Python Programming Language
0.3.1 WhyProgramming?
Whyonearthdowestartthebookwithdiscussingaprogramminglanguage?Afterall,thisisamath
(ratherthanprogramming)book!
That’strue.Butwebelievethatmanypiecesofcodeshowninthisbookwillhelpyouinmany
ways:
• Theywillshowyouarichvarietyofapplicationsofdiscretemathideasinvariousbranches
ofcomputerscience.
• Codesnippetscanserveasinteractiveexamples: youmaywanttotweakthegivenpiece
ofcode,runit,andseewhathappens.
• Bytryingtoimplementaparticularidea,youareforcedtounderstandeverysingledetail
ofit.
• Itisofteneasiertoreasonintermsofspecificobjectsinprogrammingratherthanabstract
mathematicalconcepts.
Wehavesetupeverythinginawaythatwillallowyoutorunthecodesnippetsusedinthis
bookevenifyouhavenevertriedtowriteaprogrambefore.Youdon’tevenneedtoinstallorsetup
anything:everythingcanberuninthecloud,throughyourInternetbrowser.Atthesametime,we
alsoprovideinstructionsforthosewhowouldliketolearnthebasicsofPythonwhilelearning
discretemath.
0.3.2 WhyPython?
OK,let’sdosomeprogrammingwhilelearningdiscretemath.ButwhyPythoninsteadofanyother
popularprogramminglanguage?
LetusconvinceyouthatPythonisanexcellentchoiceforourpurposes.
High-levellanguage. It is particularly easy to start using Python (even if you haven’t pro-
grammed before). The syntax is reader friendly (and close to a natural language). The
codeiscompact:mostofthepiecesofcodeinthisbookarelessthantenlineslong!
Interactivemode. Itcanbeusedinaninteractivemode(alsoknownasREPL ,forread-eval-
print loop). This allows you to talk to your computer using Python as a language: the
computerthenreadsyourinput,evaluatesit,andprintstheresult.Thisway,youworkstuff
outandgetinstantfeedbackfromthemachine.
“Batteriesincluded”. ThePythonstandardlibrary offersawiderangeoffacilities,andmany
externallibrariesareavailableaswell.Inparticular,thiswillallowustogeneratearandom
sequence,plotafunction,anddrawagraphinjustonelineofcode!
This (partly) explains why Python is often used for software prototyping, and in such areas
asmachinelearning,datascience,andwebdevelopment.
Ofcourse,advantagesalwayscomeatthecostofsomedisadvantages.Thehigh-levelnessof
Pythonmakesitlessflexibleinperformancetuning.ThisisOKforus,aswewillonlybeusing
simplesnippetsofcodewhereoptimizingisnotanissue.
0.4 Acknowledgments 9
0.3.3 HowtoCatchUpwithPython?
OK,let’stry!WheredoIstart?
Locally. ToinstallPythononyourmachine,gototheGetStarted sectionofpython.org and
followtheinstructions.IfyouarenewtoPython,weencourageyoutoinstallPyCharm
tostartworkingwithPython:this(freeofcharge)professionalIDEwillmaketheprocess
ofwritingandrunningyourcodesmootherandmoreefficient.
Inthecloud. Alternatively,youmayrunallourcodesnippetsfromyourInternetbrowser,without
installingorconfiguringanythingonyourmachine.Todothis,visittherepositorypage
andclickthebadge“OpeninColab”.Thiswillshowyoualistofnotebooksthatcanberun
inaninteractivemoderightinyourbrowser(togetherwithlinkstoatutorialonnotebooks).
0.4 Acknowledgments
Thisbookwasgreatlyimprovedbytheeffortsofalargenumberofindividualswhomweoweadebt
ofgratitude.
WethankthestudentsoftheCourseraspecializationaswellasthestudentsoftheModern
SoftwareEngineeringB.Sc.programatSt.PetersburgStateUniversityfortheircontinuousand
valuablefeedback.WealsothankJerryAllen,HuckBennett,MarieBrodsky,AnujKumarKarmakar,
andTerenceMinerbrookforcarefullyreadinganearlierdraftofthisbook.
WearegratefultoAntonKonevandDariaBorisyakforleadingthedevelopmentofinteractive
puzzles.WethankVitaliyPolshkovforreviewingourPythoncode.