The joy of finite mathematics : the language and art of math PDF
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The Joy of Finite Mathematics The Language and Art of Math The Joy of Finite Mathematics The Language and Art of Math Chris Tsokos University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620 Rebecca Wooten The Pedagogue, LLC, Developing Educational Materials, Tampa, FL AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier AcademicPressisanimprintofElsevier 125LondonWall,London,EC2Y5AS,UK 525BStreet,Suite1800,SanDiego,CA92101–4495,USA 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Copyright©2016ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyany means,electronicormechanical,includingphotocopying,recording,orany informationstorageandretrievalsystem,withoutpermissioninwritingfromthe publisher.Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’s permissionspoliciesandourarrangementswithorganizationssuchastheCopyright ClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyright bythePublisher(otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand experiencebroadenourunderstanding,changesinresearchmethods,professional practices,ormedicaltreatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledge inevaluatingandusinganyinformation,methods,compounds,orexperiments describedherein.Inusingsuchinformationormethodstheyshouldbemindfuloftheir ownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessional responsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors, oreditors,assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasa matterofproductsliability,negligenceorotherwise,orfromanyuseoroperationof anymethods,products,instructions,orideascontainedinthematerialherein. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress ISBN:978-0-12-802967-1 ForinformationonallAcademicPresspublications visitourwebsiteathttp://store.elsevier.com/ Dedications “We think faster than we speak and speak faster than we write; therefore, when creating great things we abbreviate everything. This abbreviated language is called Math.” RDW This text is dedicated to my wife Debbie and my children Matthew, Jonathan and Maria. Chris P. Tsokos This text is dedicated to my husband Dana Miller, and all my family. Rebecca D. Wooten About the Authors Chris P. Tsokos is Distinguished University Professor of Mathematics and Statistics at the University of South Florida. Dr.TsokosreceivedhisB.S.inEngineeringSciences/Mathematics,hisM.A.inMathematicsfromtheUniversityofRhode Island,andhisPh.D.inStatisticsandProbabilityfromtheUniversityofConnecticut.ProfessorTsokoshasalsoservedonthe faculties at VirginiaPolytechnic Institute and State University andthe University ofRhodeIsland. Dr. Tsokos’ research has extended into a variety of areas, including stochastic systems, statistical models, reliability analysis,ecologicalsystems,operationsresearch,timeseries,Bayesiananalysis,andmathematicalandstatisticalmodeling ofglobalwarming,bothparametricandnonparametricsurvivalanalysis,amongothers.Heistheauthorofmorethan300 research publicationsin these areas. ForthepastfouryearsProfessorTsokos’researcheffortshavebeenfocusedondevelopingprobabilisticmodels,para- metricandnonparametricstatisticalmodelsforcancerandGLOBALWARMINGdata.Specifically,hisresearchaimsare datadrivenandareorientedtowardunderstandingthebehaviorofbreast,lung,brain,andcoloncancers.Informationonthe subject matter can be found on his website. ProfessorTsokoshasmorethan300publicationsinhisresearchareasofinterest.Heistheauthorofseveralresearch monographsandbooks,includingRandomIntegralEquationswithApplicationstoLifeSciencesandEngineering,Prob- ability Distribution: An Introduction to Probability Theory with Applications, Mainstreams of Finite Mathematics with Applications,ProbabilitywiththeEssentialAnalysis,AppliedProbabilityBayesianStatisticalMethodswithApplications to Reliability, andMathematical Statistics with Applications, amongothers. Dr.Tsokosistherecipientofmanydistinguishedawardsandhonors,includingFellowoftheAmericanStatisticalAsso- ciation, USF Distinguished Scholar Award, Sigma Xi Outstanding Research Award, USF Outstanding Undergraduate Teaching Award, USF Professional Excellence Award, URI Alumni Excellence Award in Science and Technology, Pi MuEpsilon,electiontotheInternationalStatisticalInstitute,SigmaPiSigma,USFTeachingIncentiveProgram,andseveral humanitarian andphilanthropic recognitionsand awards. ProfessorTsokosisamemberofseveralacademicandprofessionalsocieties.HeisservingasHonoraryEditor,Chief- Editor,Editor orAssociate Editor ofmore than twelveacademicresearch journals. RebeccaD.WootenisAssistantProfessorofMathematicsandStatisticsattheUniversityofSouthFlorida.Shereceivedher M.A./B.A.inMathematicsandherPh.D.inStatisticsfromtheUniversityofSouthFlorida.Shehasworkedfor15yearsin teachingandhasbeenrecognizedforherexcellenceinteaching;teachingcoursessuchasLiberalArtsMath,FiniteMath- ematics, Basic Statistics, Introduction toStatistics, and AppliedStatistics Methods. HerresearchinterestsareconcentratedinAppliedStatisticswithemphasisonEnvironmentalStudies.Herresearchpub- lications span a variety of areas such as Global Warming (carbon dioxide and temperature), Atmospheric Sciences and Geography (hurricanes), Geology (volcanic ashfall),Marine Biology (red tide), among others. ProfessorWootenisextensivelyinvolvedinactivitiestoimproveeducationnotonlyinMathematicsandStatistics,but Educationingeneral.SheistheAcademicCoordinatorfortwofree-educationalassistanceprogramwhichofferopportu- nitiesforstudentstovolunteerandthelocalcommunitytogettheassistanceintheirstudiesthattheywouldotherwisebe unableto afford. xi Preface Thisbookhasbeenwrittentopresentcertainaspectsofmodernfinitemathematicsfromanelementarypointofview,with emphasisonrelevancetoreal-worldproblems.Theobjectiveistocreateapositiveattitudetowardmathematicsforthenon- science-orientatedcollegestudentandtodemonstrateitsusefulnessinsolvingproblemsthatwefrequentlyencounterinour complex society. Throughoutthetext,theaimhasbeentode-emphasizedifficulttheoreticalconcepts;thus,anintuitivetreatmentleadsto practicalapplicationsofthevarioussubjectareatopics.Webelievethatwithsuchanapproach,themoderncollegestudent will complete this course with the good feelingthat mathematics isnotonly useful but enjoyabletoworkwith. TheJoy ofFinite Mathematics hasseveral distinguishing features: Thetext hasbeen writtenfor students with only high school mathematics. Diagrams and graphs are used toillustrate mathematical conceptsorthoughts. Step-by-step directionsare given for the implementation ofmathematical methods to problem solving. Emphasis has been placed on usefulnessof mathematicsto real-world problems. To provide motivation to the reader, most chapters are preceded by a short biography of a scientist who made important contributions to the subject area under consideration. Mathematical concepts are introduced as clearly and as simply as possible, and they are followed by one or more examples asan aid tothoroughunderstanding. Eachchapterends withacompletesummarythatincludesthe definitions, properties,andrules ofthe chapter,fol- lowed by aReview Test. Eachchaptercontainsnumerouscriticalthinkingandbasisexerciseswithproblemsthatreflectonthemainstreamof the chapter. Thebookhasbeendesignedtogivetheinstructorwideflexibilityinstructuringaoneortwo-semestercourse,orafull- yearcourse.Althoughsomechaptersaredependentonotherothers,manyoptionsareallowed(seeaccompanyingdiagraph). xiii xiv Preface Chapter 1 Usefulness of Mathematics Chapter 10 Chapter 2 Chapter 3 Chapter 9 Chapter 11 Arithmetic & Logic Sets Geometry Finance Basic Algebra Chapter 4 Chapter 12 Counting Game Theory Chapter 5 Basic Probabilty Chapter 6 Chapter 7 Binomial Normal Probabilty Probabilty Chapter 8 Statistics Very Special Acknowledgment Wewishtoexpressourappreciationtothefollowingacademiceducatorsforhavingreviewedourbookandexpressedtheir opinions. An outstanding book for students to obtain basic knowledge of the usefulness of mathematics. Excellent motivation strategies throughoutthebook.Itwillinspirethestudenttolearntheimportanceofmathematics. Dr.RamKafle,DepartmentofMathematicsandStatistics,SamHoustonUniversity. Averyconstructiveandmotivatingbookoffinitemathematics.Specialemphasisontheapplicationsofmathtoreal-worldproblems. Theinteractiveapproachofpresentingtheirmaterialisexcellent.Thestudentwillacquireaverygoodunderstandingofwhatmath- ematicsisallabout. Dr.Bong-jinChoi,LinebergerComprehensiveCancerCenter,TheUniversityofNorthCarolinaatChapelHill. Thisbookisamasterfultreatmentoffinitemathematicsforundergraduatestudentswhoareafraidofmathematics.Itwillenlighten thestudentoftheinterdisciplinaryuseofmathematicsattheverybasiclevel.Thebookprovidesexcellentillustrationsoftheuseof mathematics/statisticstosolveimportantproblems. Dr.YongXu,DepartmentofMathematicsandStatistics,RadfordUniversity. TheJoyofFiniteMathematicsprovidesanexcellenttreatmentofthesubject.Uniqueemphasisontheimportanceofmathematical sciencestooursociety.Thenon-mathematics-orientedundergraduatestudentwillfindthecontentsofthebookeasytoreadandvery inspiringtolearnmoreofthesubjectmatter. Dr.K.Pokhrel,DepartmentofMathematics&ComputerSystems,MercyhurstUniversity. Thisisanexcellentbookoffinitemathematics.Itoffersajustifiable,useful,andmotivatingapproachtowhatmathematicsisallabout totheundergraduatestudentwithminimumpriorknowledgeofthesubject.Theselectionofthecontentsofthebook,examples,and exercisesisoutstanding. Dr.N.Khanal,DepartmentofMathematics,UniversityofTampa. xv Several Options for a Semester Course in Finite Mathematics Fivepossibleoptionsindesigningabasiccourseinfinitemathematicsaregivenbelow,alongwithsomeremarksforeach selection. Options 1and 2 offer adetailed coverage ofspecifictopics in math, each spanning seven chapters: Option1: ChaptersCovered Title Chapter1 TheUsefulnessofMathematics Chapter2 Logic Chapter3 Sets Chapter4 CountingTechniques Chapter5 Probability Chapter8 Statistics Chapter9 Geometry CoveringmaterialsnecessaryfortheCLAST(CollegeLevelAcademicSkillTest)exam,excludingalgebra,thesesix topicsareoftentaughtcollectively.Inadditiontothenecessaryhighschoolalgebra,thesetopicsprepareastudentwellfor the CLAST exam. Option2: ChaptersCovered Title Chapter1 TheUsefulnessofMathematics Chapter2 Logic Chapter3 Sets Chapter5 Probability Chapter6 BernoulliTrials Chapter7 TheBell-shapedCurve Chapter8 Statistics Option2providesthematerialsnecessaryforacomprehensiveunderstandingofbasicprobabilityandstatistics.This optionisabroad introduction, includingthe underlyingprobabilities necessary tocompute basic descriptivestatistics, as well asinferential statistics in terms ofinterval estimatesand tests of hypothesis. xvii xviii SeveralOptionsforaSemesterCourseinFiniteMathematics Options 3-5 offer a more detailed coverage ofspecific topics inmath, each spanning six chapters: Option 3: ChaptersCovered Title Chapter1 TheUsefulnessofMathematics Chapter3 Sets Chapter4 CountingTechniques Chapter5 Probability Chapter6 BernoulliTrials Chapter7 TheBell-shapedCurve These topics enhance the study of probability. Option 3 begins with the basic concepts of categorization into sets, countingsets,andmeasuringbasicprobabilitiesempirically.Itthencontinueswithmeasuringbasicprobabilitieshypo- theticallyusingeitherthediscretebinomialprobabilitydistribution,orthecontinuousnormalprobabilitydistribution. Option 4: ChaptersCovered Title Chapter1 TheUsefulnessofMathematics Chapter4 CountingTechniques Chapter5 Probability Chapter6 BernoulliTrials Chapter7 TheBell-shapedCurve Chapter8 Statistics Option4coversmaterialsnecessaryforthestudyofthebasicaspectsofstatistics.Thisoptionincludescountingbasic empiricalandhypotheticalprobabilitiesempirically. Italsoincludesthe basicnecessities ofstatistics,descriptivelyand inferentially,for meansand proportions. Option 5: ChaptersCovered Title Chapter1 TheUsefulnessofMathematics Chapter2 Logic Chapter3 Sets Chapter4 CountingTechniques Chapter9 Geometry Chapter11 ArithmeticandAlgebra Option5coversmaterialsnecessarytogainabasicunderstandingofthelanguageofdeterministicmath.Thisoption provides abasic understandingof logic,sets,counting, geometry, andalgebra. Note: Game theorycan be included inany scheme that includes the algebra andarithmetic.
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