The joy of finite mathematics : the language and art of math

535 Pages·2015·33.56 MB·English

by  Tsokos

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
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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.