1228AF12t1v2_sol.pdf

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Department
Mathematics
Course
Mathematics 1225A/B
Professor
Jennifer Mori
Semester
Winter

Description
SaturdayOctober132012Mathematics1228ACODE222Page1Test1PARTA18markscccc11IfnU70nA15nAB10andnAB20ndnABmarkA35B30C25D10E5ccSolutionWeseethatnAnUnA701555AndthenbypartitioningAwegetccccccccnAnABnABnABnAnAB552035cAlternativelywecouldusethegiveninformationtondnBandthusnBcccccnBnABnAB102030nBnUnB703040andalsondnABccnAnABnABnABnAnAB15105togetcnABnBnAB4053521BillisarstyearSocialSciencestudentatKingswhoistakingMath1228thistermConsiderthemarkfollowingsubsetsofWesternstudentsFcontainsallrstyearstudentsSScontainsallSocialSciencestudentsandMcontainsonlythemaincampusstudentsieMdoesnotcontainanystudentswhoareregisteredataliatecollegesatWesternKingsisoneofthealiatecollegesWhichoneofthefollowingsetsdoesnotincludeBillcccAMBMFCFSSDMFEMSScSolutionWearetoldthatBillisinFandisinSSsohesinFSSandisnotinMsohesinMSinceBillisinFhesalsointheunionofFwithanysetwhetherornothesinthatothersetsohesinMFAndsinceBillisnotinMthenhesnotintheintersectionofMwithanysetandthereforeiscinthecomplementofanysuchintersectionsohesinMFButsinceBillisinSSandthereforeisintheunionofSSwithanyothersetthenhesnotinthecomplementofanysuchunionThatisBillcisinMSSandsoheisnotinMSS31A3letterwordistobeformedusingonlyletterswhichappearinthewordcatThewordcannotstartmarkwiththeletteraandcannotendwithanaunlessthereisalreadyanainthewordThelettertcannotbeusedmorethanonceOtherwiserepetitionisallowedHowmanydierent3letterwordscanbeformedundertheserulesA27B18C12D9E6SolutionWecanconstructatreeandcounttheterminalpointsonthetreetodeterminethenumberofwordsthatcanbeformedButthattreewouldbecomplicatedtomakeonthecomputersoonlyadescriptionofthetreeisgivenhereThetreehasalevelforeachpositioninthe3letterwordiehas3levelsTheruleswemustusearethat1onlytheletterscaandtcanbeusedieeachbranchofthetreemustcorrespondtooneoftheseletters2thewordcannotstartwiththeletterasothereisnorstlevelbranchcorrespondingtoa3thewordcanonlyendinaifthereisalreadyanainthewordandsincethewordcannotstartwithathatawouldhavetobethesecondlettersoonlypathswithaonthesecondlevelbranchcanhaveaasthethirdlevelbranchand4thelettertcannotbeusedmorethanoncesonopathcanhavemorethanonetbranchTheresultingtreehasacbranchandatbranchontherstlevelGrowingfromthecbranchtherearebranchesforcandaandtbutgrowingfromtherstleveltbranchthereareonlybranchesforcand
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