Understand Mathematics, Understand Computing: Discrete Mathematics That All Computing Students Should Know

560 Pages·2020·7.525 MB·English

by  Arnold L. Rosenberg, Denis Trystram

Arnold L. Rosenberg
Denis Trystram
Understand
Mathematics,
Understand
Computing
Discrete Mathematics That All
Computing Students Should Know
Understand Mathematics, Understand Computing
Arnold L. Rosenberg • Denis Trystram
Understand Mathematics,
Understand Computing
Discrete Mathematics That All Computing
Students Should Know
Arnold L. Rosenberg Denis Trystram
College of Information Grenoble INP
and Computer Science Université Grenoble Alpes
University of Massachusetts Saint Martin d’Hères, France
Amherst, MA, USA

ISBN 978-3-030-58375-0 ISBN 978-3-030-58376-7 (eBook)
https://doi.org/10.1007/978-3-030-58376-7

© Springer Nature Switzerland AG 2020
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Addition RogerTrystram,1973
v
Tomybelovedwife,Susan,andmychildren
PaulandRachelfordecadesofpatience
duringmyextendedepisodesof“cerebral
absence.”
ToProfessorOscarZariski,whochangedmy
lifebyintroducingmeto“real”mathematics.
WithlovetomywifeCe´cileandmychildren
AliceandNoe´ whohelpmetostayconnected
tothe“real”world.
Toallpaststudentswhoinspiredthisbook.
Toallfuturestudentswhowillreadthisbook.
Contents
MANIFESTO......................................................xix
PREFACE......................................................... xxi
1 Introduction................................................... 1
1.1 WhyIsThisBookNeeded? .................................. 1
1.2 WhyIsThisBookNeeded? .................................. 2
1.3 TheStructureofThisBook .................................. 4
1.3.1 OurMainIntellectualTargets .......................... 4
1.3.2 AllocatingOurTargetstoChapters ..................... 6
1.3.2.1 Chapter2:“Doing”Mathematics ............... 6
1.3.2.2 Chapter3:SetsandTheirAlgebras ............. 6
1.3.2.3 Chapters4, 8, 10:NumbersandNumerals ...... 7
1.3.2.4 Chapters5and6:ArithmeticandSummation..... 8
1.3.2.5 Chapter7:TheVertigoofInfinity .............. 10
1.3.2.6 Chapter9:Recurrences ....................... 11
1.3.2.7 Chapter11:TheArtofCounting:Combinatorics,
withApplicationstoProbabilityandStatistics .... 13
1.3.2.8 Chapters12and13:GraphsandRelatedTopics... 15
1.4 UsingtheTextinCourses.................................... 16
1.4.1 Resources .......................................... 16
1.4.1.1 References.................................. 16
1.4.1.2 Culturalasides .............................. 18
1.4.1.3 Exercises ................................... 19
1.4.1.4 Appendices ................................. 20
1.4.2 PathsThroughtheTextforSelectedCourses ............. 20
2 “Doing”Mathematics:AToolkitforMathematicalReasoning....... 23
2.1 RigorousProofs:Theoryvs.Practice .......................... 24
2.1.1 FormalisticProofsandModernProofs .................. 26
2.1.1.1 Formalisticproofs,withanillustration .......... 26
ix
x Contents
2.1.1.2 Modernproofs,withanillustration ............. 30
2.1.1.3 Someelementsofrigorousreasoning ........... 34
A.Distinguishingnamefromobject ............ 34
B.Quantitativereasoning ..................... 34
C.Theelementsofempiricalreasoning ......... 35
2.2 OverviewofSomeMajorProofTechniques..................... 37
2.2.1 Proofby(Finite)Induction ............................ 37
2.2.1.1 Twomoresampleproofsbyinduction........... 37
2.2.1.2 Afalse“proof”byinduction:Thecriticalbase
case ....................................... 39
2.2.1.3 TheMethodofUndeterminedCoefficients ....... 40
2.2.2 ProofbyContradiction................................ 42
2.2.2.1 Theprooftechnique.......................... 42
2.2.2.2 Anothersampleproof:Thereareinfinitelymany
primenumbers .............................. 43
2.2.3 ProofsviathePigeonholePrinciple ..................... 44
2.2.3.1 Theprooftechnique.......................... 45
2.2.3.2 Sampleapplications/proofs.................... 45
2.3 NontraditionalProofStrategies ............................... 45
2.3.1 PictorialReasoning .................................. 45
2.3.1.1 Thevirtuoussideofthemethod ................ 45
2.3.1.2 Handlepictorialargumentswithcare............ 49
2.3.2 CombinatorialIntuitionandArgumentation .............. 51
2.3.2.1 Summationasaselectionproblem.............. 51
2.3.2.2 Summationasarearrangementproblem ......... 52
2.3.3 TheComputerasMathematician’sAssistant.............. 53
2.4 Exercises:Chapter2 ........................................ 54
3 SetsandTheirAlgebras:TheStemCellsofMathematics ........... 59
3.1 Introduction ............................................... 59
3.2 Sets ...................................................... 62
3.2.1 FundamentalSet-RelatedConcepts ..................... 62
3.2.2 OperationsonSets ................................... 64
3.3 StructuredSets............................................. 67
3.3.1 BinaryRelations:SetsofOrderedPairs.................. 68
3.3.2 OrderRelations...................................... 70
3.3.3 EquivalenceRelations ................................ 70
3.3.4 Functions........................................... 72
3.3.4.1 Thebasics:definitionsandgenericproperties..... 72
3.3.4.2 ⊕TheSchro¨der-BernsteinTheorem ............ 76
3.3.5 Sets,Strings,Functions:ImportantConnections .......... 78
3.4 BooleanAlgebras .......................................... 80
3.4.1 TheAlgebraofPropositionalLogic..................... 81
3.4.1.1 Logicasanalgebra .......................... 81
A.Propositions:theobjectsofthealgebra ....... 81
Contents xi
B.Logicalconnectives:thealgebra’soperations .. 82
C.Thegoalofthegame:Theorems ............. 84
3.4.1.2 Semanticcompleteness:LogicviaTruthTables... 84
A.Truth,theoremhood,andcompleteness ....... 84
B.ThecompletenessofPropositionalLogic...... 85
C.ThelawsofPropositionalLogicastheorems... 87
3.4.2 ⊕⊕APurelyAlgebraicSettingforCompleteness......... 89
3.4.3 SatisfiabilityProblemsandNP-Completeness ............ 90
3.5 Exercises:Chapter3 ........................................ 94
4 NumbersI:TheBasicsofOurNumberSystem .................... 99
4.1 IntroducingtheThreeChaptersonNumbers .................... 99
4.2 ABriefBiographyofOurNumberSystem .....................100
4.3 Integers:The“Whole”Numbers ..............................105
4.3.1 TheBasicsoftheIntegers:TheNumberLine.............106
4.3.1.1 Naturalorderingsoftheintegers................106
4.3.1.2 Theorder-relatedlawsoftheintegers ...........107
4.3.2 Divisibility:Quotients,Remainders,Divisors.............108
4.3.2.1 Euclideandivision ...........................110
4.3.2.2 Divisibility,divisors,GCDs ....................111
4.4 TheRationalNumbers ......................................113
4.4.1 TheRationals:SpecialOrderedPairsofIntegers ..........115
4.4.2 ComparingtheRationalandIntegerNumberLines........115
4.4.2.1 ComparingZandQviatheirnumber-linelaws ...116
4.4.2.2 ComparingZandQviatheircardinalities .......117
4.5 TheRealNumbers..........................................118
4.5.1 InventingtheRealNumbers ...........................118
4.5.2 DefiningtheRealNumbersviaTheirRepresentations......119
4.5.3 NotAllRealNumbersAreRational.....................120
√
4.5.3.1 Anumber-basedproofthat 2isnotrational:
√
26∈Q ....................................121
√ √
4.5.3.2 Ageometricproofthat 2isnotrational: 26∈Q 122
4.6 TheBasicsoftheComplexNumbers ..........................124
4.7 Exercises:Chapter4 ........................................125
5 Arithmetic:PuttingNumberstoWork............................127
5.1 TheBasicArithmeticOperations .............................127
5.1.1 Unary(Single-Argument)Operations ...................128
5.1.1.1 Negatingandreciprocatingnumbers ............128
5.1.1.2 Floors,ceilings,magnitudes ...................128
5.1.1.3 Factorials(ofanonnegativeinteger) ............129
5.1.2 Binary(Two-Argument)Operations.....................130
5.1.2.1 Additionandsubtraction ......................130
5.1.2.2 Multiplicationanddivision ....................131
5.1.2.3 Exponentiationandtakinglogarithms ...........133
xii Contents
5.1.2.4 BinomialcoefficientsandPascal’striangle .......135
5.1.3 RationalArithmetic:AComputationalExercise...........138
5.2 TheLawsofArithmetic,withApplications .....................139
5.2.1 TheCommutative,Associative,andDistributiveLaws .....139
5.3 PolynomialsandTheirRoots.................................140
5.3.1 UnivariatePolynomialsandTheirRoots .................142
5.3.1.1 Thed rootsofadegree-d polynomial ...........143
5.3.1.2 Solvingpolynomialsbyradicals................145
A.Solvingquadraticpolynomialsbyradicals.....146
B.Solvingcubicpolynomialsbyradicals ........148
5.3.2 BivariatePolynomials:TheBinomialTheorem ...........151
5.3.3 ⊕IntegerRootsofPolynomials:Hilbert’sTenthProblem ..153
5.4 ExponentialandLogarithmicFunctions........................155
5.4.1 BasicDefinitions ....................................155
5.4.2 LearningfromLogarithms(andExponentials)............156
5.5 ⊕PointerstoSpecializedTopics..............................159
5.5.1 NormsandMetricsforTuple-Spaces....................159
5.5.2 Edit-Distance:MeasuringClosenessinStringSpaces ......161
5.6 Exercises:Chapter5 ........................................163
6 Summation:AComplexWholefromSimpleParts .................167
6.1 TheManyFacetsofSummation ..............................167
6.2 SummingStructuredSeries ..................................171
6.2.1 ArithmeticSummationsandSeries .....................171
6.2.1.1 Generaldevelopment .........................171
6.2.1.2 Example#1:Summingthefirstnintegers........171
6.2.1.3 Example#2:Squaresassumsofoddintegers.....175
6.2.2 GeometricSumsandSeries............................179
6.2.2.1 Overviewandmainresults ....................179
6.2.2.2 Techniquesforsumminggeometricseries........180
6.2.2.3 Extendedgeometricseriesandtheirsums........185
A.ThesummationS(1)(n)=(cid:229)n ibi............186
b i=1
B.ThesummationsS(c)(n)=(cid:229)n icbi ..........188
b i=1
6.3 OnSumming“Smooth”Series ...............................190
6.3.1 ApproximateSumsviaIntegration......................190
6.3.2 SumsofFixedPowersofConsecutiveIntegers:(cid:229)ic .......193
6.3.2.1 S (n)forgeneralnonnegativerealcthpowers ....193
c
6.3.2.2 Nonnegativeintegercthpowers ................195
A.AbetterboundviatheBinomialTheorem.....195
B.Usingundeterminedcoefficientstorefinesums.197
C.Apictorialsolutionforc=2 ................199
D.⊕Semi-pictorialsolutionsforc=3..........201
6.3.2.3 S (n)forgeneralnegativecthpowers ...........207
c
A.Negativepowers−1<c<0................207
B.Negativepowersc<−1....................208