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T1_w11_sol.pdf

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Department
Mathematics
Course
Mathematics 1228A/B
Professor
Valuri
Semester
Winter

Description
SaturdayJanuary292011Mathematics1228BCODE111Page1Test1PARTA17marksc11IfU1234567A134andB356ndABmarkA23567B2567C27D56E237cccSolutionAcontainsalltheelementsofUwhicharenotinAsowehaveA2567AndABcctheintersectionofAandBcontainsonlythoseelementswhicharecontainedinbothAandBatthesametimeThereforewegetcAB256735656cccc21GiventhatnAB20nAB10nAB15andnAB40ndmarkcnABA80B65C55D40E20cccSolutionWeusetheformulanABnAnBnABOfcourseforanysetsSandTwecknowthatnSnSTnSTThatistheelementsofasetScanbepartitionedintothosewhicharealsoinsetTandthosewhicharenotsothetotalnumberofelementsinSissimplythesumcofthesetwoallinclusivenonoverlappingsubsetsofSUsingthisrstforSAandTBandnextforSBandTAandrememberingthatSTTSweseethatccccnAnABnAB154055andalsothatcnBnABnAB101525ThereforewegetcccnABnAnBnAB55251565ccAnotherApproachTondnABwecanndnAandthenaddtothatthenumberofelementscccofABwhicharenotinAwhichisonlythosewhichareinBbutnotinAiethoseinABThisgivescccccnABnAnABnABnABnAB15401065cccWecouldofcourseusenBnABinsteadofnAnABusingthesamereasoning31ThreeteamsnamedABandCaregoingtoplayatournamentasfollowsAandBwillmarkplaytherstgameTheloserofanygamedoesnotplaythenextgameThetournamentendswhenanyteamhaswontwiceHowmanypossibleoutcomesiesequencesofplayarethereA3B4C6D8E10SolutionWehave3teamsinthetournamentWearetoldwhichteamsplaytherstgameandafterthatanytimethattherehasnotyetbeenateamwhohaswon2gamestheloserofthegamesitsoutwhilethewinnerofthegameplaystheteamwhichdidntplaythatgameinthenextgameThetournamentendsassoonasanyteamhaswontwogamesWecanmodelthistournamentusingatreeThelabelson
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