




已阅读5页,还剩35页未读, 继续免费阅读
版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
P16114FIFIDIDNOTBUYALOTTERYTICKETTHISWEEK,THENIDIDNOTWINTHEMILLIONDOLLARJACKPOTONFRIDAYGIDIDNOTBUYALOTTERYTICKETTHISWEEK,ANDIDIDNOTWINTHEMILLIONDOLLARJACKPOTONFRIDAYHEITHERIDIDNOTBUYALOTTERYTICKETTHISWEEK,ORELSEIDIDBUYONEANDWONTHEMILLIONDOLLARJACKPOTONFRIDAY10ARQBPQRCRPDPQREPQRFRQP20AIFIAMTOREMEMBERTOSENDYOUTHEADDRESS,THENYOUWILLHAVETOSENDMEANEMAILMESSAGETHISHASBEENSLIGHTLYREWORDEDSOTHATTHETENSESMAKEMORESENSEBIFYOUWEREBORNINTHEUNITEDSTATES,THENYOUAREACITIZENOFTHISCOUNTRYCIFYOUKEEPYOURTEXTBOOK,THENITWILLBEAUSEFULREFERENCEINYOURFUTURECOURSESTHEWORD“THEN“ISUNDERSTOODINENGLISH,EVENIFOMITTEDDIFTHEIRGOALTENDERPLAYSWELL,THENTHEREDWINGSWILLWINTHESTANLEYCUPEIFYOUGETTHEJOB,THENYOUHADTHEBESTCREDENTIALSFIFTHEREISASTORM,THENTHEBEACHERODESGIFYOULOGONTOTHESERVER,THENYOUHAVEAVALIDPASSWORDHIFYOUDONTBEGINYOURCLIMBTOOLATE,THENYOUWILLREACHTHESUMMIT33CPQRPQPRTTTTTTFTTFTTTFFTFTTTFTFTFFTTFFFTP26128AKWAMEWILLNOTTAKEAJOBININDUSTRYANDHEWILLNOTGOTOGRADUATESCHOOLBYOSHIKODOESNTKNOWJAVAORSHEDOESNTKNOWCALCULUSCJAMESISNOTYOUNGORHEISNOTSTRONGDRITAWILLNOTMOVETOOREGONANDSHEWILLNOTMOVETOWASHINGTON10APQPPQPPQPPQQTTFTFTTFFTFTFTTTTTFFTFFTCPQPQPPQPPQQTTTTTTFFFTFTTFTFFTFT12AASSUMETHEHYPOTHESISISTRUETHENPISFALSESINCEPQISTRUE,WECONCLUDETHATQMUSTBETRUEHEREISAMORE“ALGEBRAIC“SOLUTIONPPQQPPQQPPQQPPQQPQPQTCASSUMETHEHYPOTHESISISTRUETHENPISTRUE,ANDSINCETHESECONDPARTOFTHEHYPOTHESISISTURE,WECONCLUDETHATQISALSOTRUE,ASDESIRED24PQRPQPRPQPRQRPQRTTTTTTTTTTFTFTTTTFTFTTTTTFFFFFFFFTTTTTTTFTFTTTTTFFTTTTTTFFFTTTFT30PQRPQPRQRPQPRPQPRQRTTTTTTTTTTFTFTFTTFTTTTTTTFFTFFFTFTTTTTTTFTFTTTTTFFTFTTFTFFFFTFFT51PPQPPQ977THEGRAPHISPLANARADECFB20THEGRAPHISNOTHOMEOMORPHICTOK3,3,SINCEBYREROUTINGTHEEDGEBETWEENAANDHWESEETHATITISPLANAR22REPLACEEACHVERTEXOFDEGREETWOANDITSINCIDENTEDGESBYASINGLEEDGETHENTHERESULTISK3,3THEPARTSAREA,E,IANDC,G,KTHEREFORETHISGRAPHISHOMEOMORPHICTOK3,323THEGRAPHISPLANAR25THEGRAPHISNOTPLANAR9833AFBCED839210417TIMESLOT1MATH115,MATH185TIMESLOT2MATH116,CS473TIMESLOT3MATH195,CS101TIMESLOT4CS102TIMESLOT5CS273P46133ATRUEBFALSECFALSEDFALSE5ATHEREISASTUDENTWHOSPENDSMORETHAN5HOURSEVERYWEEKDAYINCLASSBEVERYSTUDENTSPENDSMORETHAN5HOURSEVERYWEEKDAYINCLASSCTHEREISASTUDENTWHODOESNOTSPENDMORETHAN5HOURSEVERYWEEKDAYINCLASSDNOSTUDENTSPENDSMORETHAN5HOURSEVERYWEEKDAYINCLASS9AXPXQXBXPXQXCXPXQXDXPXQX16ATRUEBFALSECTRUEDFALSE24LETCXBETHEPROPOSITIONALFUNCTION“XISINYOURCLASS”AXPXANDXCXPX,WHEREPXIS“XHASACELLULARPHONE”BXFXANDXCXFX,WHEREFXIS“XHASSEENAFOREIGNMOVIE”CXSXANDXCXSX,WHERESXIS“XCANSWIM”DXEXANDXCXEX,WHEREEXIS“XCANSOLVEQUADRATICEQUATIONS”EXRXANDXCXRX,WHERERXIS“XWANTSTOBERICH”62AXPXSXBXRXSXCXQXPXDXQXRXEYESIFXISONEOFMYPOULTRY,THENHEISADUCKBYPARTC,HENCENOTWILLINGTOWALTZPARTASINCEOFFICERSAREALWAYSWILLINGTOWALTZPARTB,XISNOTANOFFICERP591412DXCX,BOBHXYIXXYIYKXYIXCX,YNXYZXYCX,ZCY,Z14AXHX,WHEREHXIS“XCANSPEAKHINDI”ANDTHEUNIVERSEOFTHEDISCOURSECONSISTSOFALLSTUDENTSINTHISCLASSBXYPX,Y,WHEREPX,YIS“XPLAYSY”ANDTHEUNIVERSEOFTHEDISCOURSEFORXCONSISTSOFALLSTUDENTSINTHISCLASS,ANDTHEUNIVERSEOFTHEDISCOURSEFORYCONSISTSOFALLSPORTSCXAXHX,WHEREAXIS“XHASVISITEDALASKA”,HXIS“XHASVISITEDHAWAII”ANDTHEUNIVERSEOFTHEDISCOURSEFORXCONSISTSOFALLSTUDENTSINTHISCLASSDXYLX,Y,WHERELX,YIS“XHASLEARNEDPROGRAMMINGLANGUAGEY”ANDTHEUNIVERSEOFTHEDISCOURSEFORXCONSISTSOFALLSTUDENTSINTHISCLASS,ANDTHEUNIVERSEOFTHEDISCOURSEFORYCONSISTSOFALLPROGRAMMINGLANGUAGESEXZYQY,ZPX,Y,WHEREPX,YIS“XHASTAKENCOURSEY”,QY,ZIS“COURSEYISOFFEREDBYDEPARTMENTZ”,ANDTHEUNIVERSEOFTHEDISCOURSEFORXCONSISTSOFALLSTUDENTSINTHISCLASS,THEUNIVERSEOFTHEDISCOURSEFORYCONSISTSOFALLCOURSESINTHISSCHOOL,ANDTHEUNIVERSEOFTHEDISCOURSEFORZCONSISTSOFALLDEPARTMENTSINTHISSCHOOLFXYZXYPX,YXYZPX,Z,WHEREPX,YIS“XANDYGREWUPINTHESAMETOWN”ANDTHEUNIVERSEOFTHEDISCOURSEFORX,Y,ZCONSISTSOFALLSTUDENTSINTHISCLASSGXYZCX,YGY,Z,WHERECX,YIS“XHASCHATTEDWITHY”,GY,ZIS“YISINCHATGROUPZ”,THEUNIVERSEOFTHEDISCOURSEFORX,YCONSISTSOFALLSTUDENTSINTHISCLASS,ANDTHEUNIVERSEOFTHEDISCOURSEFORZCONSISTSOFALLCHATGROUPINTHISCLASS24ATHEREISANADDITIVEIDENTITYFORTHEREALNUMBERSDTHEPRODUCTOFTWONONZERONUMBERSISNONZEROFORTHEREALNUMBERS38BTHEREARENOSTUDENTSINTHISCLASSWHOHAVENEVERSEENACOMPUTERDTHEREARENOSTUDENTSINTHISCLASSWHOHAVETAKENBEENINATLEASTONEROOMOFEVERYBUILDINGONCAMPUS151RQPPQRRQPPQRQPPQRPQRQPQRPPQR30,1,2,4,5,6,72P726LETRBETHEPROPOSITION“ITRAINS“,LETFBETHEPROPOSITION“ITISFOGGY“,LETSBETHEPROPOSITION“THESAILINGRACEWILLBEHELD“,LETLBETHEPROPOSITION“THELIFESAVINGDEMONSTRATIONWILLGOON“,ANDLETTBETHEPROPOSITION“THETROPHYWILLBEAWARDED“WEAREGIVENPREMISESRFSL,ST,ANDTWEWANTTOCONCLUDERWESETUPTHEPROOFINTWOCOLUMNS,WITHREASONSNOTETHATITISVALIDTOREPLACESUBEXPRESSIONSBYOTHEREXPRESSIONSLOGICALLYEQUIVALENTTOTHEMSTEPREASON1THYPOTHESIS2STHYPOTHESIS3SMODUSTOLLENSUSINGSTEPS1AND24RFSLHYPOTHESIS5SLRFCONTRAPOSITIVEOFSTEP46SLRFDEMORGANSLAWANDDOUBLENEGATIVE7SLADDITION,USINGSTEP38RFMODUSPONENSUSINGSTEP6AND79RSIMPLIFICATIONUSINGSTEP812FIRST,USINGTHECONCLUSIONOFEXERCISE11,WESHOULDSHOWTHATTHEARGUMENTFORMWITHPREMISESPTRS,QUT,UP,S,Q,ANDCONCLUSIONRISVALIDTHEN,WEUSERULESOFINFERENCEFROMTABLE1STEPREASON1QPREMISE2QUTPREMISE3UTMODUSPONENSUSINGSTEPS1AND24USIMPLIFICATIONUSINGSTEP35UPPREMISE6PMODUSPONENSUSINGSTEPS3AND47TSIMPLIFICATIONUSINGSTEP38PTCONJUNCTIONUSINGSTEPS6AND79PTRSPREMISE10RSMODUSPONENSUSINGSTEPS8AND911SPREMISE12RDISJUNCTIVESYLLOGISMUSINGSTEPS10AND1114BLETRXBE“XISONEOFTHEFIVEROOMMATES,”DXBE“XHASTAKENACOURSEINDISCRETEMATHEMATICS,”ANDAXBE“XCANTAKEACOURSEINALGORITHMS”THEPREMISESAREXRXDX,XDXAXANDRMELISSAUSINGTHEFIRSTPREMISEANDUNIVERSALINSTANTIATION,RMELISSADMELISSAFOLLOWSUSINGTHETHIRDPREMISEANDMODUSPONENS,DMELISSAFOLLOWSUSINGTHESECONDPREMISEANDUNIVERSALINSTANTIATION,AMELISSAFOLLOWSSODOTHEOTHERROOMMATESDLETCXBE“XISINTHECLASS,”FXBE“XHASBEENTOFRANCE,”ANDLXBE“XHASVISITEDLOUVRE”THEPREMISESAREXCXFXANDXFXLXFROMTHEFIRSTPREMISEANDEXISTENTIALINSTANTIATIONIMPLYTHATCYFYFORAPARTICULARPERSONYUSINGSIMPLIFICATION,FYFOLLOWSUSINGTHESECONDPREMISEANDUNIVERSALINSTANTIATIONFYLYFOLLOWSUSINGMODUSPONENS,LYFOLLOWSUSINGEXISTENTIALGENERALIZATION,XCXLXFOLLOWS24THEERRORSOCCURINSTEPS3,5AND7FORSTEPS3AND5,WECANNOTASSUME,ASISBEINGDONEHERE,THATTHECTHATMAKESPXTRUEISTHESAMEASTHECTHATMAKESQXTRUEATTHESAMETIMEFORSTEP7,ITISNOTACONJUNCTIONANDTHEREISNOSUCHDISJUNCTIONRULE29STEPREASON1XPXPREMISE2PCEXISTENTIALINSTANTIATIONFROM13XPXQXPREMISE4PCQCUNIVERSALINSTANTIATIONFROM35QCDISJUNCTIVESYLLOGISMFROM2AND46XQXSXPREMISE7QCSCUNIVERSALINSTANTIATIONFROM68SCDISJUNCTIVESYLLOGISMFROM5AND79XRXSXPREMISE10RCSCUNIVERSALINSTANTIATIONFROM911RCMODUSTOLLENSFROM8AND1012XRXEXISTENTIALGENERALIZATIONFROM11P861637SUPPOSETHATP1P4P2P5P3P1TOPROVETHATONEOFTHESEPROPOSITIONSIMPLIESANYOFTHEOTHERS,JUSTUSEHYPOTHETICALSYLLOGISMREPEATEDLYP1031713ATHISSTATEMENTASSERTSTHEEXISTENCEOFXWITHACERTAINPROPERTYIFWELETYX,THENWESEETHATPXISTRUEIFYISANYTHINGOTHERTHANX,THENPXISNOTTRUETHUS,XISTHEUNIQUEELEMENTTHATMAKESPTRUEBTHEFIRSTCLAUSEHERESAYSTHATTHEREISANELEMENTTHATMAKESPTRUETHESECONDCLAUSESAYSTHATWHENEVERTWOELEMENTSBOTHMAKEPTRUE,THEYAREINFACTTHESAMEELEMENTTOGETHERTHESESAYTHATPISSATISFIEDBYEXACTLYONEELEMENTCTHISSTATEMENTASSERTSTHEEXISTENCEOFANXTHATMAKESPTRUEANDHASTHEFURTHERPROPERTYTHATWHENEVERWEFINDANELEMENTTHATMAKESPTRUE,THATELEMENTISXINOTHERWORDS,XISTHEUNIQUEELEMENTTHATMAKESPTRUEP120219TTFTTF16SINCETHEEMPTYSETISASUBSETOFEVERYSET,WEJUSTNEEDTOTAKEASETBTHATCONTAINSASANELEMENTTHUSWECANLETAANDBASTHESIMPLESTEXAMPLE20THEUNIONOFTHESETSINTHEPOWERSETOFASETXMUSTBEEXACTLYXINOTHERWORDS,WECANRECOVERXFROMITSPOWERSET,UNIQUELYTHEREFORETHEANSWERISYES22ATHEPOWERSETOFEVERYSETINCLUDESATLEASTTHEEMPTYSET,SOTHEPOWERSETCANNOTBEEMPTYTHUSISNOTTHEPOWERSETOFANYSETBTHISISTHEPOWERSETOFACTHISSETHASTHREEELEMENTSSINCE3ISNOTAPOWEROF2,THISSETCANNOTBETHEPOWERSETOFANYSETDTHISISTHEPOWERSETOFA,B28AA,X,0,A,X,1,A,Y,0,A,Y,1,B,X,0,B,X,1,B,Y,0,B,Y,1,C,X,0,C,X,1,C,Y,0,C,Y,1C0,A,X,0,A,Y,0,B,X,0,B,Y,0,C,X,0,C,Y,1,A,X,1,A,Y,1,B,X,1,B,Y,1,C,X,1,C,YP1302214SINCEAABAB,WECONCLUDETHATA1,5,7,83,6,91,3,5,6,7,8,9SIMILARLYBBAAB2,103,6,92,3,6,9,1024FIRSTSUPPOSEXISINTHELEFTHANDSIDETHENXMUSTBEINABUTINNEITHERBNORCTHUSXAC,BUTXBC,SOXISINTHERIGHTHANDSIDENEXTSUPPOSETHATXISINTHERIGHTHANDSIDETHUSXMUSTBEINACANDNOTINBCTHEFIRSTOFTHESEIMPLIESTHATXAANDXCBUTNOWITMUSTALSOBETHECASETHATXB,SINCEOTHERWISEWEWOULDHAVEXBCTHUSWEHAVESHOWNTHATXISINABUTINNEITHERBNORC,WHICHIMPLIESTHATXISINTHELEFTHANDSIDE40THISISANIDENTITYEACHSIDECONSISTSOFTHOSETHINGSTHATAREINANODDNUMBEROFTHESETSA,B,ANDCP1472335ATHISREALLYHASTWOPARTSFIRSTSUPPOSETHATBISINFSTTHUSBFAFORSOMEASTEITHERAS,INWHICHCASEBFS,ORAT,INWHICHCASEBFTTHUSINEITHERCASEBFSFTTHISSHOWSTHATFSTFSFT,CONVERSELY,SUPPOSEBFSFTTHENEITHERBFSORBFTTHISMEANSEITHERTHATBFAFORSOMEASORTHATBFAFORSOMEATINEITHERCASE,BFAFORSOMEAST,SOBFSTTHISSHOWSTHATFSFTFST,ANDOURPROOFISCOMPLETEBSUPPOSEBFSTTHENBFAFORSOMEASTTHISIMPLIESTHATASANDAT,SOWEHAVEBFSANDBFTTHEREFOREBFSFT,ASDESIRED52INSOMESENSETHISQUESTIONISITSOWNANSWERTHENUMBEROFINTEGERSBETWEENAANDB,INCLUSIVE,ISTHENUMBEROFINTEGERSBETWEENAANDB,INCLUSIVEPRESUMABLYWESEEKANEXPRESSINVOLVINGA,B,ANDTHEFLOORAND/ORCEILINGFUNCTIONTOANSWERTHISQUESTIONIFWEROUNDAUPANDROUNDBDOWNTOINTEGERS,THENWEWILLBELOOKINGATTHESMALLESTANDLARGESTINTEGERSJUSTINSIDETHERANGEOFTHEINTEGERSWEWANTTOCOUNT,RESPECTIVELYTHESEVALUESAREOFCOURSEAND,RESPECTIVELYTHENTHEANSWERISB1JUSTTHINKOFCOUNTINGALLTHEINTEGERSBETWEENTHESETWOVALUES,ABINCLUDINGBOTHENDSIFAROWOFFENCEPOSTSONEFOOTAPARTEXTENDSFORKFEET,THENTHEREAREK1FENCEPOSTSNOTETHATTHISEVENWORKSWHEN,FOREXAMPLE,A03ANDB07P1622434ATHISISCOUNTABLETHEINTEGERSINTHESETARE1,2,4,5,7,ANDSOONWECANLISTTHESENUMBERSINTHEORDER1,1,2,2,4,4,THEREBYESTABLISHINGTHEDESIREDCORRESPONDENCEINOTHERWORDS,THECORRESPONDENCEISGIVENBY11,21,32,42,54,ANDSOONBTHISISSIMILARTOPARTAWECANSIMPLYLISTTHEELEMENTSOFTHESETINORDEROFINCREASINGABSOLUTEVALUE,LISTINGEACHPOSITIVETERMBEFOREITSCORRESPONDINGNEGATIVE5,5,10,10,15,15,20,20,30,30,40,40,45,45,50,50,CTHISISCOUNTABLEBUTALITTLETRICKYWECANARRANGETHENUMBERSINA2DIMENSIONALTABLEASFOLLOWS11011011101111011111111111111111111111111111111111111111111111111111111111111111111DTHISSETISNOTCOUNTABLEWECANPROVEITBYTHESAMEDIAGONALIZATIONARGUMENTASWASUSEDTOPROVETHATTHESETOFALLREALSISUNCOUNTABLEINEXAMPLE21ALLWENEEDTODOISCHOOSEDI1WHENDII9ANDCHOOSEDI9WHENDII1ORDIIISBLANKIFTHEDECIMALEXPANSIONISFINITE46WEKNOWFROMEXAMPLE21THATTHESETOFREALNUMBERSBETWEEN0AND1ISUNCOUNTABLELETUSASSOCIATETOEACHREALNUMBERINTHISRANGEINCLUDING0BUTEXCLUDING1AFUNCTIONFROMTHESETOFPOSITIVEINTEGERSTOTHESET0,1,2,3,4,5,6,7,8,9ASFOLLOWSIFXISAREALNUMBERWHOSEDECIMALREPRESENTATIONIS0D1D2D3WITHAMBIGUITYRESOLVEDBYFORBIDDINGTHEDECIMALTOENDWITHANINFINITESTRINGOF9S,THENWEASSOCIATETOXTHEFUNCTIONWHOSERULEISGIVENBYFNDNCLEARLYTHISISAONETOONEFUNCTIONFROMTHESETOFREALNUMBERSBETWEEN0AND1ANDASUBSETOFTHESETOFALLFUNCTIONSFROMTHESETOFPOSITIVEINTEGERSTHESET0,1,2,3,4,5,6,7,8,9TWODIFFERENTREALNUMBERSMUSTHAVEDIFFERENTDECIMALREPRESENTATIONS,SOTHECORRESPONDINGFUNCTIONSAREDIFFERENTAFEWFUNCTIONSARELEFTOUT,BECAUSEOFFORBIDDINGREPRESENTATIONSSUCHAS0239999SINCETHESETOFREALNUMBERSBETWEEN0AND1ISUNCOUNTABLE,THESUBSETOFFUNCTIONSWEHAVEASSOCIATEDWITHTHEMMUSTBEUNCOUNTABLEBUTTHESETOFALLSUCHFUNCTIONSHASATLEASTTHISCARDINALITY,SOIT,TOO,MUSTBEUNCOUNTABLEP191321THECHOICESOFCANDKARENOTUNIQUEAYESC1,K10BYESC4,K7CNODYESC5,K1EYESC1,K0FYESC1,K29X24X173X3FORALLX17,SOX24X17ISOX3,WITHWITNESSESC3,K17HOWEVER,IFX3WEREOX24X17,THENX3CX24X173CX2FORSOMEC,FORALLSUFFICIENTLYLARGEX,WHICHIMPLIESTHATX3C,FORALLSUFFICIENTLYLARGEX,WHICHISIMPOSSIBLEP2093419ANOBNOCYESDNO31AGRQRWSDVVJRBQBABGCNFFTBCQXUXMAHJJZXP2183513AYESBNOCYESDYES17A2B4C12P2804122ALITTLECOMPUTATIONCONVINCESUSTHATTHEANSWERISTHATN2NFORN0,1,ANDALLN4CLEARLYTHEINEQUALITYDOESNTHOLDFORN2ORN3WEWILLPROVEBYMATHEMATICALINDUCTIONTHATTHEINEQUALITYHOLDSFORALLN4THEBASECASEISCLEAR,SINCE1624NOWSUPPOSETHATN2NFORAGIVENN4WEMUSTSHOWTHATN12N1EXPANDINGTHELEFTHANDSIDE,APPLYINGTHEINDUCTIVEHYPOTHESIS,ANDTHENINVOKINGSOMEVALIDBOUNDSSHOWSTHISN22N1N2N1N2N1N3NNNNNNNN1NN1P2934231ASSUMETHATTHEWELLORDERINGPROPERTYHOLDSSUPPOSETHATP1ISTRUEANDTHATTHECONDITIONALSTATEMENTP1P2PNPN1ISTRUEFOREVERYPOSITIVEINTEGERNLETSBETHESETOFPOSITIVEINTEGERSNFORWHICHPNISFALSEWEWILLSHOWSASSUMETHATS,THENBYTHEWELLORDERINGPROPERTYTHEREISALEASTINTEGERMINSWEKNOWTHATMCANNOTBE1BECAUSEP1ISTRUEBECAUSENMISTHELEASTINTEGERSUCHTHATPNISFALSE,P1,P2,PM1ARETRUE,ANDM11BECAUSEP1P2PM1PMISTRUE,ITFOLLOWSTHATPMMUSTALSOBETRUE,WHICHISACONTRADICTIONHENCE,SP3084310THEBASECASEISTHATSM0MTHERECURSIVEPARTISTHATSMN1ISTHESUCCESSOROFSMNIE,SMN112THEBASECASEN1ISCLEAR,SINCEF12F1F21ASSUMETHEINDUCTIVEHYPOTHESISTHENF12F22FN2FN12FN12FNFN1FN1FN1FNFN1FN2,ASDESIRED31IFXISASETORVARIABLEREPRESENTINGSET,THENXISWELLFORMEDFORMULAIFXANDYAREALLWELLFORMEDFORMULAS,THEN,XY,XYANDXYAREALLWELLFORMEDXFORMULAS50LETPNBE“A1,N2N”BASICSTEPP1ISTRUE,BECAUSEP1A1,1221INDCUTIVESTEPASSUMETHATPMISTRUE,THATISA1,M2MANDM1THENPM1A1,M1A0,A1,MA0,2M22M2M1SOA1,N2NWHENEVERN159BNOTWELLDEFINEDF2ISNOTDEFINEDSINCEF0ISNTALSO,F2ISAMBIGUOUSDNOTWELLDEFINEDTHEDEFINITIONISAMBIGUOUSABOUTN1P344513A104B512WEUSETHESUMRULE,ADDINGTHENUMBEROFBITSTRINGSOFEACHLENGTHUPTO6IFWEINCLUDETHEEMPTYSTRING,THENWEGET2021222324252627112720AEVERYSEVENTHNUMBERISDIVISIBLEBY7THEREFORETHEREARE999/7142SUCHNUMBERSNOTETHATWEUSETHEFLOORFUNCTION,BECAUSETHEKTHMULTIPLEOF7DOESNOTOCCURUNTILTHENUMBER7KHASBEENREACHEDBFORSOLVINGTHISPARTANDTHENEXTFOURPARTS,WENEEDTOUSETHEPRINCIPLEOFINCLUSIONEXCLUSIONJUSTASINPARTA,THEREARE999/1190NUMBERSINOURRANGEDIVISIBLEBY11,ANDTHEREARE999/7712NUMBERSINOURRANGEDIVISIBLEBYBOTH7AND11THEMULTIPLESOF77ARETHENUMBERSWESEEKIFWETAKETHESE12NUMBERSAREAWAYFROMTHE142NUMBERSDIVISIBLEBY7,WESEETHATTHEREARE130NUMBERSINOURRANGEDIVISIBLEBY7BUTNOTBY11CASEXPLAINEDINPARTB,THEANSWERIS12DBYTHEPRINCIPLEOFINCLUSIONEXCLUSION,THEANSWER,USINGTHEDATAFROMPARTB,IS1429012220EIFWESUBTRACTFROMTHEANSWERTOPARTDTHENUMBEROFNUMBERSDIVISIBLEBYNEITHEROFTHEMSOTHEANSWERIS22012208FIFWESUBTRACTTHEANSWERTOPARTDFROMTHETOTALNUMBEROFPOSITIVEINTEGERSLESSTHAN1000,WEWILLHAVETHENUMBEROFNUMBERSDIVISIBLEBYEXACTLYONEOFTHEMSOTHEANSWERIS999220779GIFWEASSUMETHATNUMBERSAREWRITTENWITHOUTLEADING0S,THENWESHOULDBREAKTHEPROBLEMDOWNINTOTHREECASESONEDIGITNUMBERS,TWODIGITNUMBERSCLEARLYTHEREARE9ONEDIGITNUMBERS,ANDEACHOFTHEMHASDISTINCTDIGITSTHEREARE90TWODIGITNUMBERS10THROUGH99,ANDALLBUT9OFTHEMHAVEDISTINCTDIGITSANALTERNATIVEWAYTOCOMPUTETHISISTONOTETHATTHEFIRSTDIGITMUSTBE1THROUGH99CHOICESANDTHESECONDDIGITMUSTBESOMETHINGDIFFERENTFROMTHEFIRSTDIGIT9CHOICESOUTOFTHE10POSSIBLEDIGITS,SOBYTHEPRODUCTRULE,WEGET9981CHOICESINALLTHISAPPROACHALSOTELLSUSTHATTHEREARE998648THREEDIGITNUMBERSWITHDISTINCTDIGITSAGAIN,WORKFROMLEFTTORIGHTINTHEONESPLACE,ONE8DIGITSARELEFTTOCHOOSEFROM80THEFINALANSWERIS981648738HITTURNSOUTTOBEEASIERTOCOUNTTHEODDNUMBERSWITHDISTINCTDIGITSANDSUBTRACTFROMOURANSWERTOPARTG,SOLETUSPROCEEDTHATWAYTHEREARE5ODDONEDIGITNUMBERSFORTWODIGITNUMBERS,FIRSTCHOOSETHEONEDIGIT5CHOICES,THENCHOOSETHETENSDIGITS8CHOICES,SINCENEITHERTHEONESDIGITVALUENOT0ISAVAILABLETHEREFORETHEREARE40SUCHTWODIGITNUMBERSNOTETHATTHISISNOTEXACTLYHALFOF81FORTHETHREEDIGITNUMBERS,FIRSTCHOOSETHEONESDIGIT5CHOICES,THENTHEHUNDREDSDIGIT8CHOICES,THENTHETENSDIGIT8CHOICES,GIVINGUS320INALLSOTHEREARE540320365ODDNUMBERSWITHDISTINCTDIGITSTHUSTHEFINALANSWERIS73836537335A若N1,为2;若N2,为2若N3,为0B对于N1,为;若N1,为1;NC2N1注N可映射到0,1两种可能44FIRSTWECOUNTTHENUMBEROFBITSTRINGSOFLENGTH10THATCONTAINFIVECONSECUTIVE0SWEWILLBASETHECOUNTONWHERETHESTRINGOFFIVEORMORECONSECUTIVE0SSTARTSIFITSTARTSINTHEFIRSTBIT,THENTHEFIRSTFIVEBITSAREALL0S,BUTTHEREISFREECHOICEFORTHELASTFIVEBITSTHEREFORETHEREARE2532SUCHSTRINGSIFITSTARTSINTHESECONDBIT,THENTHEFIRSTBITMUSTBEA1,THENEXTFIVEBITSAREALL0S,BUTTHEREISFREECHOICEFORTHELASTFOURBITSTHEREFORETHEREARE2416SUCHSTRINGSIFITSTARTSINTHETHIRDBIT,THENTHESECONDBITMUSTBEA1BUTTHEFIRSTBITANDTHELASTTHREEBITSAREARBITRARYTHEREFORETHEREARE2416SUCHSTRINGSSIMILARLY,THEREARE16SUCHSTRINGSTHATHAVETHECONSECUTIVE0SSTARTINGINEACHOFPOSITIONSFOUR,FIVE,ANDSIXTHISGIVESUSATOTALOF32516112STRINGSTHATCONTAINFIVECONSECUTIVE0SSYMMETRICALLY,THEREARE112STRINGSTHATCONTAINFIVECONSECUTIVE1SCLEARLYTHEREAREEXACTLYTWOSTRINGSTHATCONTAINBOTH0000011111,1111100000THEREFOREBYTHEINCLUSIONEXCLUSIONPRINCIPLE,THEANSWERIS2112222252WEDRAWTHETREE,WITHITSROOTATTHETOPWESHOWABRANCHFOREACHOFTHEPOSSIBILITIES0AND1,FOREACHBITINORDER,EXCEPTTHATWEDONOTALLOWTHREECONSECUTIVE0SSINCETHEREARE13LEAVES,THEANSWERIS1310FIRSTBIT1010SECONDBIT1010101THIRDBIT1010101101010FOURTHBITP353526THEREAREONLYDPOSSIBLEREMAINDERSWHENANINTEGERISDIVIDEDBYD,NAMELY0,1,D1BYTHEPIGEONHOLEPRINCIPLE,IFWEHAVED1REMAINDERS,THENATLEASTTWOMUSTBETHESAME10THEMIDPOINTOFTHESEGMENTWHOSEENDPOINTSAREA,BANDC,DISAC/2,BD/2WEARECONCERNEDONLYWITHINTEGERVALUESOFTHEORIGINALCOORDINATESCLEARLYTHECOORDINATESOFTHESEFRACTIONSWILLBEINTEGERSASWELLIFANDONLYIFAANDCHAVETHESAMEPARITYBOTHODDORBOTHEVENANDBANDDHAVETHESAMEPARITYTHUSWHATMATTERSINTHISPROBLEMISTHEPARITIESOFCOORDINATESTHEREAREFOURPOSSIBLEPAIRSOFPARITIESODD,ODD,ODD,EVEN,EVEN,EVENANDEVEN,ODDSINCEWEAREGIVENFIVEPOINTS,THEPIGEONHOLEPRINCIPLEGUARANTEESTHATATLEASTTWOOFTHEMWILLHAVETHESAMEPAIROFPARTIESTHEMIDPOINTOFTHESEGMENTJOININGTHESETWOPOINTSWILLTHEREFOREHAVEINTEGERCOORDINATES38ATBTCT1A13ANAN1AN2AN3BTHEINITIALCONDITIONSAREA01,A11,ANDA22,SINCETHEREISONEWAYTOCLIMBNOSTAIRSDONOTHING,CLEARLYONLYONEWAYTOCLIMBONESTAIR,ANDTWOWAYSTOCLIMBTWOSTAIRSONESTEPTWICEORTWOSTEPSATONCENOTETHATTHERECURRENCERELATIONISTHESAMEASTHATFOREXERCISE25CEACHTERMINOURSEQUENCEANISTHESUMOFTHEPREVIOUSTHREETERMS,SOTHESEQUENCEBEGINSA11,A22,A34,A47,A513,A624,A744,A881THUSAPERSONCANCLIMBAFLIGHTOF8STAIRSIN81WAYSUNDERTHERESTRICTIONSINTHISPROBLEM42LETANBETHENUMBEROFCOVERINGSAWEFOLLOWTHEHINTIFTHERIGHTMOSTDOMINOISPOSITIONEDVERTICALLY,THENWEHAVEACOVERINGOFTHELEFTMOSTN1COLUMNS,ANDTHISCANBEDONEINAN1WAYSIFTHERIGHTMOSTDOMINOISPOSITIONEDHORIZONTALLY,THENTHEREMUSTBEANOTHERDOMINODIRECTLYBENEATHIT,ANDTHESETOGETHERCOVERTHELASTTWOCOLUMNSTHEFIRSTN2COLUMNSTHEREFOREWILLNEEDTOCONTAINACOVERINGBYDOMINOS,ANDTHISCANBEDONEINAN2WAYSTHUSWEOBTAINTHEFIBONACCIRECURRENCEANAN1AN2BCLEARLYA11ANDA22CTHESEQUENCEWEOBTAINISJ
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 初中物理教学全套
- 湖南省岳阳市一中下学期2025年高三二诊模拟考试英语试卷含解析
- 北京市北方交通大学附属中学2025届高三第一次模拟考试英语试卷含解析
- 2025届陕西省汉中市汉台区县高三第三次测评英语试卷含答案
- 天津市红桥区2024-2025学年高二下学期4月期中考试 化学试题(原卷版+解析版)
- 精神康复中的游戏治疗技术考核试卷
- 天津市宝坻区2024-2025学年高三高考适应性模拟押题测试(一)化学试题含解析
- 四川省绵阳市江油市2025年初三5月第四次模拟考试数学试题含解析
- 江南影视艺术职业学院《企业形象策划》2023-2024学年第一学期期末试卷
- 山西省运城市重点中学2024-2025学年高考考前质量检测试题(三)历史试题含解析
- (完整word版)电梯曳引系统设计-毕业设计
- 2023建筑施工承插型套扣式钢管脚手架安全技术规程
- 组织供应,运输,售后服务方案
- (完整版)各档口单品菜品毛利率核算表
- 2023年云南省昆明市中考作文真题解析及欣赏:坚持的力量
- SMC电磁阀的选型手册
- 2023年江苏泰州市第四人民医院招考聘用高层次人才11人模拟备考试卷(共1000题含答案解析)
- 工会换届选举请示样式
- 七年级音乐上册 《青少年管弦乐队指南》教学课件
- 新中国史智慧树知到答案章节测试2023年
- 员工面试登记表通用模板
评论
0/150
提交评论