Discrete Mathematics: An Open Introduction, 3rd Edition PDF
Preview Discrete Mathematics: An Open Introduction, 3rd Edition
Discrete Mathematics An Open Introduction Oscar Levin 3rd Edition Discrete Mathematics An Open Introduction Oscar Levin 3rd Edition OscarLevin SchoolofMathematicalScience UniversityofNorthernColorado Greeley,Co80639 [email protected] http://math.oscarlevin.com/ © 2013-2021byOscarLevin ThisworkislicensedundertheCreativeCommonsAttribution-ShareAlike 4.0InternationalLicense. Toviewacopyofthislicense,visit http://creativecommons.org/licenses/by-sa/4.0/. 3rdEdition 5thPrinting: 1/7/2021 ISBN:978-1792901690 Acurrentversioncanalwaysbefoundforfreeat http://discrete.openmathbooks.org/ Coverimage: TilingwithFibonacciandPascal. ForMadelineandTeagan Acknowledgements Thisbookwouldnotexistifnotfor“DiscreteandCombinatorialMathe- matics”byRichardGrasslandTabithaMingus. ItisthebookIlearned discretemathoutof,andtaughtoutofthesemesterbeforeIbeganwriting this text. I wanted to maintain the inquiry based feel of their book but update, expand and rearrange some of the material. Some of the best expositionandexerciseshereweregraciouslydonatedfromthissource. ThankstoAleesSeehausenwhoco-taughttheDiscreteMathematics course with me in 2015 and helped develop many of the Investigate! activitiesandotherproblemscurrentlyusedinthetext. Shealsooffered manysuggestionsforimprovementoftheexpositorytext,forwhichIam quitegrateful. ThanksalsotoKatieMorrison,NateEldredgeandRichard Grassl(again)fortheirsuggestionsafterusingpartsofthistextintheir classes. Whileoddsarethattherearestillerrorsandtyposinthecurrentbook, there are many fewer thanks to the work of Michelle Morgan over the summerof2016. Thebookisnowavailableinaninteractiveonlineformat,andthisis entirelythankstotheworkofRobBeezer,DavidFarmer,andAlexJordan alongwiththerestoftheparticipantsofthepretext-supportgroup. Finally,athankyoutothenumerousstudentswhohavepointedout typos and made suggestions over the years and a thanks in advance to thosewhowilldosointhefuture. v vi Preface Thistextaimstogiveanintroductiontoselecttopicsindiscretemathe- maticsatalevelappropriateforfirstorsecondyearundergraduatemath majors,especiallythosewhointendtoteachmiddleandhighschoolmath- ematics. The bookbeganas aset ofnotesfor theDiscrete Mathematics courseattheUniversityofNorthernColorado. Thiscourseservesbothas asurveyofthetopicsindiscretemathandasthe“bridge”courseformath majors,asUNCdoesnotofferaseparate“introductiontoproofs”course. Most students who take the course plan to teach, although there are a handfulofstudentswhowillgoontograduateschoolorstudyapplied mathorcomputerscience. Forthesestudentsthecurrenttexthopefully is still of interest, but the intent is not to provide a solid mathematical foundationforcomputerscience,unlikethemajorityoftextbooksonthe subject. Another difference between this text and most other discrete math books is that this book is intended to be used in a class taught using problemorientedorinquirybasedmethods. WhenIteachtheclass,Iwill assignsectionsforreadingafterfirstintroducingtheminclassbyusing amixofgroupworkandclassdiscussiononafewinterestingproblems. Thetextismeanttoconsolidatewhatwediscoverinclassandserveasa referenceforstudentsastheymastertheconceptsandtechniquescovered intheunit. None-the-less,everyattempthasbeenmadetomakethetext sufficientforselfstudyaswell,inawaythathopefullymimicsaninquiry basedclassroom. The topics covered in this text were chosen to match the needs of thestudentsIteachatUNC.Themainareasofstudyarecombinatorics, sequences,logicandproofs,andgraphtheory,inthatorder. Inductionis coveredattheendofthechapteronsequences. Mostdiscretebooksput logicfirstasapreliminary,whichcertainlyhasitsadvantages. However,I wantedtodiscusslogicandproofstogether,andfoundthatdoingboth of these before anything else was overwhelming for my students given thattheydidn’tyethavecontextofotherproblemsinthesubject. Also, after spending a couple weeks on proofs, we would hardly use that at allwhencoveringcombinatorics,somuchoftheprogresswemadewas quicklylost. Instead,thereisashortintroductionsectiononmathematical statements,whichshouldprovideenoughcommonlanguagetodiscuss thelogicalcontentofcombinatoricsandsequences. Dependingonthespeedoftheclass,itmightbepossibletoinclude additionalmaterial. InpastsemestersIhaveincludedgeneratingfunctions (aftersequences)andsomebasicnumbertheory(eitherafterthelogicand vii viii proofschapterorattheveryendofthecourse). Theseadditionaltopics arecoveredinthelastchapter. WhileI(currently)believethisselectionandorderoftopicsisoptimal, you should feel free to skip around to what interests you. There are occasionally examples and exercises that rely on earlier material, but I havetriedtokeepthesetoaminimumandusuallycaneitherbeskipped orunderstoodwithouttoomuchadditionalstudy. Ifyouareaninstructor, feelfreetoedittheLATEXorPreTeXtsourcetofityourneeds. Improvementstothe3rdEdition. Inadditiontolotsofminorcorrections,bothtotypographicalandmath- ematical errors, this third edition includes a few major improvements, including: • Morethan100newexercises,bringingthetotalto473. Theselection of which exercises have solutions has also been improved, which shouldmakethetextmoreusefulforinstructorswhowanttoassign homeworkfromthebook. • Anewsectioninontreesinthegraphtheorychapter. • Substantialimprovementtotheexpositioninchapter0,especially thesectiononfunctions. • The interactive online version of the book has added interactivity. Currently,manyoftheexercisesaredisplayedasWeBWorKproblems, allowingreaderstoenteranswerstoverifytheyarecorrect. Thepreviouseditions(2ndedition,releasedinAugust2016,andthe Fall 2015 edition) will still be available for instructors who wish to use thoseversionsduetofamiliarity. Myhopeistocontinueimprovingthebook,releasinganewedition eachspringintimeforfalladoptions. Theseneweditionswillincorporate additionsandcorrectionssuggestedbyinstructorsandstudentswhouse thetexttheprevioussemesters. ThusIencourageyoutosendalongany suggestionsandcommentsasyouhavethem. OscarLevin,Ph.D. UniversityofNorthernColorado,2019
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