Table Of Content

Springer Undergraduate Mathematics SerÎes Springer-Verlag London Ltd. AdvisoryBoard P.j.Cameron QueenMaryandWestfieldCollege MAJ.Chaplain UniversityofDundee K.Erdmann OxfordUniversity L.c'G.Rogers UniversityofBath E.Silli OxfordUniversity j.F.Toland UniversityofBath Otherbooksinthisseries AnalyticMethodsforPartialDifferentialEquationsG.Evans,I.Blackledge,P.Yardley Applied GeometryforComputerGraphicsand CADD.Marsh BasicLinearAlgebra T.S.BlythandE.F.Robertson BasicStochasticProcesses Z.BrzezniakandT.Zastawniak ElementaryDifferentialGeometry A.Pressley ElementaryNumberTheory G.A.lonesandI.M.[ones ElementsofLogicviaNumbersandSets D.L.[ohnson Groups, RingsandFields D.A.R.Wallace HyperbolicGeometryI.W.Anderson Informationand CodingTheory G.A.[onesandI.M.lones IntroductiontoLaplaceTransformsand FourierSeries P.P.G.Dyke IntroductiontoRingTheory P.M.Cohn IntroductoryMathematics:AlgebraandAnalysis G.Smith IntroductoryMathematics:ApplicationsandMethods G.S.Marshall LinearFunctionalAnalysis RP.RynneandM.A.Youngson Measure,Integraland Probability M.CapinksiandE.Kopp MultivariateCalculusand Geometry S.Dineen NumericalMethodsforPartialDifferentialEquationsG.Evans,J.Blackledge,P.Yardley Sets,Logicand Categories P.Cameron TopicsinGroupTheory G.SmithandO.Tabachnikova TopologiesandUniformities I.M.lames VectorCalculus P.e.Matthews Ian Anderson AFirst Course in Discrete Mathematics With 63Figures , Springer lan Anderson, MA, PhD Department of Mathematics, U niversity of Glasgow, University Gardens, GlasgowG12 8QW, UK CiJl't'T illlHrT"ti{)n elemenrs reproduced by kind pPTmi,I.\iorl n/ Artc..:h Systems, lnc., Publishers of the GAUSS Mathematl..::al and Slatisu..:al Srstem, 23804 S.E. KCllt-Kilnglcy Rnad. Maple Villle)', WA 980~8, USA. Tel: (206) 4J2 -7855 Fax (206) 432 -78.n email: info@aptech .. .::omURL:www.apteeh .. .::om Ameri~'<lll Statistica! Association: Chance VolS No 1. 1995 article b)' KS and KW Heiner 'Tree Rings ofthe Northern Shawangunks' page 32 tig 2 Srringer-Verliig: Mathematica in Educatian and Research Vol4 Issue.3 1995 afticle by Roman E Maeder. Bcatrice Amrhein and Oliver Gloor 'ILlustrdted Mathematks: Visualization ofMathematical Objects' page 9 fig Il, originaU)' published as a CD ROM 'lllustrated Mathematics' by TELOS:ISBN 978-1-))5233-236-5, German edition by Birknauser: ISBN 978-1-))5233-236-5 Millnen1J!i..:a in Edu..:~tion and Researcn Vol4 Issue 3 1995 artide br Richard J Gaylord and Kazume Nisrudatl' 'Trafti..: Engineering with Cellular Automata' page:l5 fig 2. Mathematk~ in Edueation and Researen Voi 5 Issue 21996 artide by Mîchael Tron 'The Implidtization of a Trefoil Knot' page 14. !\:l~tht'matica in Education and Researeh Vals Issue 21996 article by Lee de Cola 'Coms, Trees, Bars and Bells: Simulation of the Binoerual Pro cess' page 19 fig ,:1, Mathematiea in Edueatian and Resear..:h ValS [ssue 219% artic1e by Richard Gaylord and Kazume Nishidate 'Contagious Sl'rcading' page 33 tlg 1. Mathcmatica in Edueatian and Resear..:h Voi 5 [ssue 2 1996 article by Joe Buhler and Stan Wagon 'Secrets of the MaJdullg Constant' rage 50 fig l. Springer Undergraduate Mathematics Series ISSN 1615-2085 British Library Cataloguing in Publiealion Data Anderson, lan, 1942- A first eourse in discrete rnathemalies. -(Springn unJcrgraduate mathematics series) 1. Computer sciencc -!vlathcmatics 2. Cumbinatorial analysis 1. Titli.' 510 Lihrary ofCongress Cataloging-in-Puhlieatinn Data Anucrsun, lan, 1942- A fir.~t course in discrete mathematies Ilan Anderson. p. (.m. -- (Springer undergraduate mathematics serics) InduJC's bibliographical references and index. ISBN 978-1-85233-236-5 ISBN 978-0-85729-315-2 (eBook) DOI 10.1007/978-0-85729-315-2 1. 1\1athematics. 2. Computer SciencC' -Matht:rnatics. 1. Title. II. Series. 'lA.lY.2.AS33 2000 SHI--ddl 00-063702 ArdeI rrom aoy fair dcaling for the purposcs of research or private study, or criticism nf review, aS permitted umler the Copyright, Designs and Patents Act 1988, this publicat ion may onIy be reproduced. stored or transmitted, in any farm or by any mcans, with th(' prior permission in writing ofthe publishers, OT in the case of reprographic r"production in accordancc wilh the terms of Iicences issued by the Copyright Licensing Agency. FnquiriC's concerning reproductian outside thosc terms should be sent to the publishcrs © Springer-Verlag London 2002 Originally published by Springer-Verlag London Limited in 2002 MyCopy version ofthe original edition 2002 The USi.: uf rcgistcred names, trademarks etc. in this publication does not imply. cven in the absencc of a specific stalcmcnt, thal such namcs arc exempt rrom the relevant laws and regulations and therefore frec for general use. l'he publishcr mah's no reprcscntation, expres!> or implicd, with regard to Ih(' ac(uracy of thc information cnntdJOcd in this book and cannut accept any legal responsibility or liability for any efrors or omissions that may !le lIIadC'. Type<;et ling: Camera rcady by the aulhor amI. Michael Mackey 12/3R30-54321 Printcd on acid-frec paper SPIN 10887527 www.springer.com/mycopy Preface Thisaddition to theSUMSseriesoftextbooksisanintroduction tovariousas pectsofdiscretemathematics.It isintendedas atextbookwhich could beused at undergraduatelevel,probablyin the second year ofanEnglish undergradu ate mathematics course.Some textbookson discrete mathematicsare written primarilyfor computing science students,but the present book isintended for students followinga mathematicscourse.Theplaceofdiscretemathematicsin the undergraduate curriculum is now fairly well established,and it is certain that its place in the curriculum willbe maintained in the third millennium. Discrete mathematics has several aspects. One fundamental part is enumeration,thestudy ofcounting arrangements ofvarious types.Wemight count thenumber ofwaysofchoosing sixlottery numbersfrom 1,2,...,49,or the number of spanning trees in a complete graph,or the number of waysof arranging 16teamsinto four groups offour.Wedevelop methods ofcounting which can deal with such problems. Next, graph theory can be used to model a variety of situations - road systems, chemical molecules, timetables for examinations. We introduce the basictypesofgraphandgivesome indicationofwhat theimportantproperties arethat a graph might possess. Thethird areaofdiscretemathematicstobediscussed inthisbook isthatof configurationsorarrangements.Latin squaresarearrangementsofsymbols in a particular way; such arrangementscan be used to construct experimental designs,magicsquaresandtournamentdesigns.Thisleadsusontohave alook at block designs, which were discussed extensively by statisticians as well as mathematicians dueto their usefulnessin thedesign ofexperiments.The book closes with a briefintroduction tothe ideas behind error-correctingcodes. The reader does not require a great deal of technical knowledge to be able tocope with thecontentsofthe book. Aknowledgeofthe method ofproofby induction,anacquaintancewith theelementsofmatrixtheoryandofarithmetic modulo n,a familiarity with geometricseries and a certain clarity ofthought v vi Preface should see the reader through. Often the main problem encountered by the reader is not in the depth of the argument,but in looking at the problem in the "right way".Facility inthiscomesofcourse withpractice. Each chapterends with a good number ofexamples.Hints and solutions to most ofthesearegivenat theendofthehook.Theexamplesareamixtureof fairly straightforward applications ofthe ideas of the chapter and more chal lenging problems which are of interest in themselves or areof use later on in thehook. My hopeis that this text will providethebasisfor a first course in discrete mathematics. Obviously thechoiceofmaterial for such a course is dependent ontheinterestsoftheteacher,butthereshould beenoughtopicsheretoenable an appropriate choice to be made. The text has been influenced in countless ways by the many texts that have appeared over the years, but ultimately it is determined by my own preferences, likes and dislikes, and by my own experienceofteachingdiscretemathematicsatdifferent levelsovermanyyears, from masterclassesfor 14-year-oldsto final year honourscourses. Iwouldliketothank theSpringerstafffortheir encouragement to writethis bookandfortheirhelpin its production.ThanksisalsoduetoGail Henryfor convertingmymanuscriptinto a~TEX file,and toMarkThomson forreading andcommentingonmanyofthechapters. University ofGlasgow,June2000 Contents 1. Counting and Binomial Coefficients......................... 1 1.1 BasicPrinciples .......................................... 1 1.2 Factorials..................................... 2 1.3 Selections 3 1.4 Binomial Coefficients and Pascal's Triangle .................. 6 1.5 Selections with Repetitions ........... 10 1.6 A Useful MatrixInversion ............................... 13 2. Recurrence ................................................ 19 2.1 Some Examples ......................................... 19 2.2 TheAuxiliary EquationMethod............................ 23 2.3 Generating Functions ..................................... 26 2.4 Derangements............................................ 28 2.5 Sorting Algorithms ................................ 32 2.6 Catalan Numbers......................................... 34 3. Introduction to Graphs ..................................... 43 3.1 The Conceptofa Graph 43 3.2 Pathsin Graphs.......................................... 46 3.3 Trees ................................................... 47 3.4 SpanningTrees........................................... 50 3.5 Bipartite Graphs ......................................... 52 3.6 Planarity................................................ 54 3.7 Polyhedra ............................................... 60 4. Travelling Round a Graph .................................. 69 4.1 HamiltonianGraphs 69 4.2 Planarityand Hamiltonian Graphs 71 4.3 TheTravellingSalesmanProblem 74 vii viii Contents 4.4 GrayCodes.............................................. 76 4.5 Eulerian Graphs.... 78 4.6 Eulerian Digraphs .. 81 5. PartitionsandColourings........................... 89 5.1 Partitions ofa Set ........................................ 89 5.2 Stirling Numbers... ......................... 91 5.3 CountingFunctions....................................... 94 5.4 Vertex Colourings ofGraphs ............................... 96 5.5 Edge ColouringsofGraphs 99 6. The Inclusion-Exclusion Principle 107 6.1 The Principle 107 6.2 Counting Surjections 112 6.3 Counting Labelled Trees 113 6.4 Scrabble 114 6.5 The MenageProblem 115 7. Latin Squares and Hall's Theorem 121 7.1 Latin Squaresand Orthogonality 121 7.2 MagicSquares 125 7.3 SystemsofDistinct Representatives 127 7.4 From Latin Squares to Affine Planes 131 8. Schedules and I-Factorisations 137 8.1 The Circle Method 137 8.2 BipartiteTournaments and l-Factorisations of Kn,n 142 8.3 Tournamentsfrom Orthogonal LatinSquares 145 9. Introduction to Designs 149 9.1 Balanced Incomplet.e Block Designs 149 9.2 Resolvable Designs 156 9.3 Finite ProjectivePlanes 159 9.4 Hadamard Matricesand Designs 161 9.5 Difference Met.hods ....... . 165 9.6 HadamardMatricesand Codes 167 Appendix 179 Solutions . . 183 Further Reading. . 1% Bibliography.. ..............................................197 Index 199 1 Counting and Binomial Coefficients In this chapter we introduce the basic counting metbods, the factorial func tion andthe binomialcoefficients.Theseareoffundamentalimportancetothe subject matter ofsubsequent chapters.Westart with twobasicprinciples. 1.1BasicPrinciples (a) The multiplication principle. Suppose that an activity consists of k stages,andthattheithstagecanbecarriedoutinQ;different ways,irrespective ofhowthe otherstagesarecarriedout.Then thewholeactivitycan becarried out in QIQ2 •.•Qk ways. Example1.1 A restaurant serves three types ofstarter, six main courses and five desserts. So a three-coursemealcan bechosenin 3x 6x 5= 90ways. (b) The addition principle. IfAI,... ,A are pairwise disjoint sets (i.e. k A;nA = 0whereveri l'j),thenthe number ofelements in their union is j k IAIu..·uAkl=IAII+..+IAkl=LIA;.I i=l Example 1.2 In theaboveexample,how manydifferent two-coursemeals (including a main course) are there? I. Anderson, A First Course in Discrete Mathematics © Springer-Verlag London 2002 2 Discrete Mathematics Solution There are two types of two-course mealto consider.Let Al denote the set of meals consisting of a starter and a main course, and let A2 denote the set of mealsconsisting ofa maincourse and a dessert.Then the required number is IAII+IA21 (by theaddition principle) (3 x6)+(6 x5) (by the multiplication principle) 48. 1.2Factorials How many ways are there ofplacing a,band cin a row? There are six ways, namely abc,acb,bac,bca,cab,cba. Note that there are three choices for the first place,then twofor the second, and then just one for the third; so by the multiplication principle there are 3 x 2x 1= 6 possibleorderings.In general,ifwedefine n! ("n factorial") by n! = n(l1- 1)(n - 2)... 2.1 then we have Theorem 1.1 The number ofwaysofplacing11objects in orderisn1. Example1.3 Fourpeople,A,B,C,D,formacommittee.Oneistobepresident,onesecretary, one treasurer, and one social convener. In how many ways can the posts be assigned? Solution Think offirst choosing a president,then a secretary,and soon.There4!= 24 possiblechoices. The value ofn! gets big very quickly: 5!= 120, 1O! = 3628800, 50!~3.04x 1064. This isanexampleofwhat is called combinatorial explosion:the number of different arrangementsof11objects getshugeas11increases.Theenormoussize of11! lies behind what is knownas the travellingsalesman problem which will