ICS 6B (Winter '11) HW Solutions

ICS 6B (Winter '11) HW Solutions - CHAPTER 1 The...

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Unformatted text preview: CHAPTER 1 The Foundations: Logic and Proofs SECTION 1.1 Propositional Logic 2. Proposition must hmciauly defined tluth values. so a proposition must he ndecluatiw sentence with no fluminbla. a) This is not a Brownian; it‘s acommand. b) This is not a pmpuition; it‘soquation. c) Thishapmpodtionthatiafabmasmmwhohnbeentohlaineknwn. d) This isnota proposition; haunt]: value dependson the vain-90f z. a) This in: proposition that b tube. 0 This is not. a proposition; its truth value depends on the value of n. 4. a) Idid not buy olottay ticket that week. b) Either] bought a lottery ticket thisweek or [in the inclusive scam] I won the million dollar jndipot on hid“. c)[flbouginabtmryfichetthhweeknhenimnthemflhondolhrjackpotonfiiday. d) lbmngiflabtmryticketthhmkandlmthemifliondoflujukpotonfikhy. o) lbwght alotten'ticlaet this mhiiaud oulyiilmnthemiiibndollnrjadwot on Nday. Olfldiduotbuyubttuytichetthismk.thenldidnotwinthemilliondollujackpotonfi'idw. g)ld|d notbuyalottu'ytichet thiweehmndIdidnotwlnthemillbndollarjackpotonmday. h) Bithcrldidnotbuyalotterytichatth'nmhmeheldklbuyonenndmnthemilliondollmjukpoton Friday. 5-1.0. a) The election is not decided. :1 lane dectionlsdeddedmrthevom hmbecnwunted. :¢)Theelectionionotdecided.mdthemhmboeacounwd. '2 dilfuwmtahavebcenoounud.theutheeleaionisdecided. _'« .)“themtuhavcnotbeeucoumd.thantheeiectionlsmtdecided. jiouuueuiocuonunocdeemedmmmmhmmmm. anmmbmummyuummmmmm. munmmhavanotbeenmumcd.mdsetheehctionismtdecidedandfltevotahmbeencounmd. ‘hflwmmubletoiuoorpormthepumthemhyusingthemrdseiihcrmddoe. "Milnvathciiu.thenwuminthefinnlm. '3! YWdonotmhutheflnalmmiiandonlyiiyoupmthecoum Humminthefinaimmdimyoudomtmtheooum. if. Mthoflumrmiuthefimlmmmtpusthcmum. Wthamthutifwuhmthcfluthmwudomtpamthecoumeorthecuethatiiwumiu ," mmfluhmmtkmhrbothithundemtood}. 1:“ Mm‘hluandmimthefimluunmrmudonotmhthcfiunlmmmddoputhem. 2 Chapter] WWW.” 10. a) rA-q b) pAqAr c)r—op d)pl\-1q/\r o) (pAq)—or 0ro—o(qu) 13. a)Thi|hT-o’l‘,vhichhtme. h)'l‘hilhTo-ol’,vhichlnfnhe. c)Th‘thHF,whichhtmL-. d)ThlshP6-o’l‘.whkhbf|br. u. n)'l‘hhiaP—ol?.whldtitme. h) Thbhl‘dl’.llltkhiatrm. c) Thls‘nT—ol’.whirhhluhe. d) Thisla'l‘w'l‘. whichhtme. lo. a) Theemploymmkingthi-mueatmldhehappyfltheappliumknowhothotumlanmmoothb belarlyan itmluive or. 18. a) The Way conditlon is the comhuion: "you get promoted. then you ml: the bo-‘n our. b) It Inc winds are [run the muth. than them will he a spring tluw. c) The. suflirlmt mlltion k the hypothuh: I! you bought the amputet la: than a not on then the mummy is good. d) If Willy chem. tlwn he ms caught. 4!) The “only if” mnditkm is the coodmlm: If you new the Website, than you must pay a subscription I'm. 0 I! you know ttu- right people, than you will he oledal. 3) "Carol houathJhethQ-geum-k. 20. I) If I am In remember to send mu the “than, than )ou will have to send me an o-null w. (This In- bmndightlymdulmthatthetemmuhmm.) b) If you wane born in the Unlted States. than you an! a citizen of thls «nutty. e) lt‘youg-nhejothenywhadthebutcmkmhh g) lfyouhgontothethmyouhnwnv-lldpmd. b) You will behixmedll'nndnnlylfmmodtln- wmmythy. (It-minds bettulntllllordn; It mid he logically equlnleut to cute this as “You read the mumps awry day If and only It you will be, {Augeethcurluldifmdonlyifhnisnotlm . ulmhomonhmltwfllsuowhonight. Contrapaitiva: Ifldommynhomthmlcwfll ;’ Converse: WWIwmthMithamnylmdny. Court-spasm“): thumldonotgo mMJthmtnmnnywmmerday. Inverse: Whenamithnotamnnydqdeomtgototlnbmch. Guam: lflaleapuntilnoonfihenlstqeduphm. Conn-apoalth‘e: IfldonotalaepuntilnoonJlnnI . ammupm. lnwnc: lfldon'tmyuplate,thenldon‘tsleepuntllnoon. “Auuthmblewmneadrmmhmmnmmbn 1.09:4 b)2a=s c)2‘=u d)25=32 ‘ Iboonatmctdnetmth table Soracomponnd pmpositlon,weworkfmmtheinsideom. Inwhmae,wewlll Mthointormediatentepo. In part (d),formmple.weflmtoonumcuhe truth ublasfiorpAqand for qu nndoombimtlnmtogettlnuuth tablefor (pAq) -o (quJ. Papal-(1(a) and (Muchavcthe Mable [column thmfiorputh). column burlnrpanw». g 1 P-‘T pH" '1' F F F F T T P mm (c) and (.1)me the followlng table. 2!: We M9 paw) (pAq~(ch T T T T F T T F T F F T F T T F T T F F' F F F T Forpart(o)nhmthefllawln¢mble. u 1 1-H? P"? mm T T P P T F T F F T F F F T T T F F F F T T T T Fbrpart(f)whavethefiwllowlngtable p q 3 Put: P°""q vulva-(paw) T T F T P T T F T F T '1‘ F T F F T T F F T T F T have the following table (column two [or put (a), column four fat put 02)]. g per 1 row T F P T F F T T Pbrpnm(c)and(d)wehavetheblbwingtable(columnsfiwmdnlx). ullmm TT F r T r T? F T F T PT T F F T 1?? T T T F m m m W Chapter! explicitly showing the negation of . deflldtiom In Section L2). V e 0 AF??? \al. r 9 A TPTFTFFF V b $ r 0 v vTTTT WTTTTTTTF w, b. q KTTTTTTPF MTFPT r TFTPTPTF QTTFFTTFF MPTTF PTTTTPFPF GTFTP LTTFF 32. For part: (a) and (b), I“: have T A q A 0 r V M. TTTFTFTF [w a. ATTPFFFFP r TFTFTPTF VTTPFTTFF PTTTTFFFF For parts (c) and (d), we have TFPFFFFF T V flTTFTFrFT A mTTFPFFFP w A «FTFTFTPF v 0 a. WTITTTTTFP IFTFTFTPT r TFTFTFTF QTTFFTTFF P TTTTFFFF Finally, for puts (a) and (f) we have 84. Thiatimethetruthtubleneodaz‘nlfi ma. wmmm TTTT T T 'r TTTF T T F TTFT T F T TTFP T F ’l‘ TFTT F T T TFTP F T P TPFT F T' T TFFF F T F FTTT T T T FTTF T T F FTFT T F T FTFF T F T FFTT T T T FFTF '1‘ T F PFFT T P T FFFF T F T V a) Stone the condition istrue, the daemon: lacxecuu-dmo z isincmnented and now has the value 2. "' b) Sinmthecmditionishbgthemntcnmisnotmutodmo: hoot lncmmuntednndnawstlllhasthe vuluol. c) Since the condltlonlstrue. the statement ismted,sor lslncmnntodondmhuthe value 2. d) Sincetheoondltlonbfoln.thoatmttilnotmted.flozlsnotincramenudandnowstillhuthe value 1. e) Sinoethecmulitionistmewlmitheomntemdwnmx- l).theatatemmtismuted.oox is incremented and now huthavalue 2. (It In lnelevont that thcoondition i now false.) . a) llOlIlA(01011Vl1011)=11MA11011=11W b) (01111A10101)V0 I!!!) = 00101 V0 1W=0 1101 c) (01010911011)®010w =1 1111100111!) - 111111 (I) (11011V01010)A(10001V11011)-11011A11011=11011 - The truth value of ”Pied and John are happy” '8 mln(0.8,0.4) = 0.4. The truth valueof “Nelther M not John is happy" 13 mln(0.2.0.6) = 0.2. since this statement means “Fwd is not happy. and John is not happy." and we computed the ttuth values of the two pmposltlons In this conjunction In Exmclse 35. .Thisconnotbenpmposltlon,becuneltmnnothnveotmthmluo. Wflltmmflmitwoukl he truly cum-ting that it 15 false. a oontmdictlon: on tho othot hand If it we (also. then its ligation that it is {also must be false, 90’ that it would be true—again a cootmdlctlon. Thus th'n string of letters, while appeafingtobeapmpooitiondsinfwtmnimleu. No. Thialaoclaalcalpundox. (Wawillmethemnleplouounlnwhnt followsaaulmlngthatnomtolklng oboutlnalenahavingtheitbeuth Manna annulus that ollmcn havobcialhalr. lfwemtflct onmlvuto Mandollowl'emnleborhemthen thcbotbercouldhofemnlewithnocontndlctlon.) [fa-thalamus athed.whomldlhovetheharbet? [fthehorbenhovodhimoelffihenhewuldbcviolntingthemlethat hashmoulythooepeoplowhodonotuhmthemselm Ontheothcrhonddfhodomnotshmhlmnell. theotheruleaoysthathommtshmhlmmlf. Neitherlnpomlbleaothmoconhenosudnbubet. a) Iftheexplomr (om,sothatourpmooum wlllnotnetconhnedhemtlnconnlhalswlllhenmlc) encoununatmth-tellar.thenhewlllhooesuymm “no“ tohetquation. lfaheeooomollthcntho 48. 52. 64. honeatanawertoherqueationia'yea,'eohewilliieandamwer“no.' Tlunaevu'ybocbwiilamwer'm- thequation.nndtheexplomrwillhmmwaywdeterminewhidttypeofcannibalahehspeahingm, b) Theremeevemlpouihleoonectam Oneiathebllowingqumtion: “Ifiweretoaakyouif alwayatoldthetruth,wuidyouenythatwudidr Thenil‘theoannibaliantruthteller.hewiliamm (tmthflnllyLVhileifhehallar.thm.sinoeinhcthewouidhaveanidthathedidtellthetruthifquutM. hewillnowiieandanswerno. I) “But” means “and": rA -vp. b) ‘Whenetu’ means “if”: (Hip) -.q. o) Acmbelngdeniediathenegationofqmowehnve wr—o-wq. d) The hypothes’u is a conjunction: (13A 1-) -¢ q. Wewritctlmsymholically: u—a-u,a—oa,-u—owu. Nmethatwemnmahalltheooncluslontmeby maldrgafidae.atrue.andufalne. Mnfltlmmcmnotaooeutlnfilemmmeyunmm tiles. and thesyatem knot being upgraded. then all the conditional statesnecnaaretrue. Thu “team'- imminent. Thiaqstem inconsistent. Wanna L. Q, N. and B tostand Iotthebasicpropoaitlonshere. ”1110me is locked.” “New messages will be queued," “The system is functioning normally.” and “New W will beamttotheumagebufl'w." leapectlveb'. Then thefimapecificationsare «L —oQ. -‘Lo-o N. -Q—o B. nL—aB.and -B. Il'wwantcmnistenchhenwehad betterhave B falscinorderthat HB betme. This requires that both I. and Q be two. by them conditional statements that have 8 as their consequence The first conditional statement therefore is of the form F —. T. which '3 true. Finally. the biconditional «L H N can he anti-Bod by taking N to be folnc. Thus this act of apuiMutiuus h annulment. Note that that: is jlfl this one satisfying truth assignment. This is similar to Example 17, about universities in New Mexico. ’l'b search tor hiking In West Virginia. we could enter WEST AND VIRGINIA AND HIKING. If we enter (VIRGINIA AND HIKING) NOT WEST. then we’ll get websites about hiking in Virginia but not in West Virginia. except for sites that happen to use the wood 'waat" in a difl'erent context (e.g.. “F‘nllow the stream vat until )ou come to a clearing“). . If A isnlmight. then In“. atatement. that both ol'them are knights b true, and both will be telling the truth. But tlm Ialmpousible‘ became B loosening otherw'ne (that A i aknave). If A he know, then B'- eaartloniatruc.aohemuatbeaknight,nnd A’saaaertionlslhlne.uitahouldbe. Thuawoconciudethat A Baltnaveand B haknlght. . “k! can draw no conclusions. A knight will declare himself to be a knight. telling the tmth. A known will lie nnduaettthathebnknight.Slmeoveryonewllleayflamahnightfweeandctumimnothing. .n) Welookatthethmepoeaibilitieaofwhotheinnooatmenmighthe. “Smithlnd-lonesueinnooent (and therefore telling the truth), that we get. an immediate contradiction. since Smith said that Jena m a friend ofCoopet. but Jones said that he did not earn know Cooper. [Lions and Williams are the innomnt tnnbmlkrgthenweapingetacmmudiotion.dmehnumynthatbedidnotknowCoopei-andmom~ of town. but Williams says hem Jones with Cooper (presumably in town. and presumably if“ was with him. then helmet! him). Thenfaeltmust bothecaaethatSmlthandWlllhmmicllingthetruth. Their mmmcnudonotoOMndicteachotha. BanedonWflllamn'ntatenmt.weknaI'thntJoneathing,mhe aaidthathedidnmkmeooperwheninfactl-ewaawithhim. Wloneahthemurderer. Section 1.2 Propositional Eqrrivaiencas 7 64. b)Thisisjurnliinepart(a).cetceptthatwearenotwhiaheadoi’timethatoneoi’themenlsguiity. Can none of them heguiity? Um. then they are all telling the truth. but this 'n impouihle. because as “just saw.aorneoi'thestaternentsarcooutradictory. Canrnore thanoneoi'thembeguilty? If. loreuampie,they areallguilty. then their statements give usaoinformation. Sothat iscertainly possible. This inbrmation iscnough to determine the entire system. Let each letter stand for the statement that the person whose name begins with that letter is chatting. Then the giwn information no he expressed syndrolicallyasl’ollows: -K--H.R—oaV.~R-oV.A—oR,V-bK,K-oV.H—oA,H—oK. Notethatwewereabletoconvertalloithesestatementaintocomlitionalstatementa.lnwhatlollcwswewili sometimes mob-meddle contrapoaitlveao‘thue conditional statementcaswcll. First mwoaethat H is true. Thcnitioliowsthat A and K aretrue,wheneeitfollowsthat R and V aretrne. But R impllathat beaheaowegetaoontradlction. Therefore”mustbefalae.Fromth’nitlolkwsthatKistrue;quice VimandflwrefomkbiabaasisA.Wecannowdieckthatthisuignrnentleadstoatrnevaluelor eaeheonditional statement. Sow. concludethat Kevin and Vijayarechattingbutlioather, Randy. and Abby arenot. Note that Diana's statement is merely that she didn‘t do it. a) John did it. There are four cases to consider. if Alice is the sole truth-teller, then Carlos did It; but this means that John is telling the truth. a contradiction. if John is the solo truth-teller. then Diana must be lying. so she did it. but then Carlos is telling the truth. a contradiction. If Carlos is the sole truth-teller, then Diana did it, but that mains John truthful. again a contradiction. So the only posibility is that Diana is the sole truth-teller. Thh means that John is lying when he denied it, so he did it. Note that in this case both Alice and Carlos are indeed lying. h) Again there are bur cases to consider. Since Carlos and Diana are making contradictory statements, the liar must be one ofthern (we could hate used this approach in part (a) as well]. Therefore Alice is telling the truth. a) Carlos did it. Note that John and Diana are telling the truth as well here. and it is Carlos who is lying. ...
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