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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 vain90f 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)ldd notbuyalottu'ytichet thiweehmndIdidnotwlnthemillbndollarjackpotonmday. h) Bithcrldidnotbuyalotterytichatth'nmhmeheldklbuyonenndmnthemilliondollmjukpoton
Friday. 51.0. 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) rAq b) pAqAr c)r—op d)pl\1q/\r o) (pAq)—or 0ro—o(qu) 13. a)ThihTo’l‘,vhichhtme.
h)'l‘hilhTool’,vhichlnfnhe.
c)Th‘thHF,whichhtmL.
d)ThlshP6o’l‘.whkhbfbr. u. n)'l‘hhiaP—ol?.whldtitme.
h) Thbhl‘dl’.llltkhiatrm.
c) Thls‘nT—ol’.whirhhluhe.
d) Thisla'l‘w'l‘. whichhtme. lo. a) Theemploymmkingthimueatmldhehappyﬂtheappliumknowhothotumlanmmoothb
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 houathJhethQgeumk. 20. I) If I am In remember to send mu the “than, than )ou will have to send me an onull w. (This In
bmndightlymdulmthatthetemmuhmm.)
b) If you wane born in the Unlted States. than you an! a citizen of thls «nutty. e) lt‘yougnhejothenywhadthebutcmkmhh g) lfyouhgontothethmyouhnwnvlldpmd. b) You will behixmedll'nndnnlylfmmodtln wmmythy. (Itminds 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. Courtspasm“): thumldonotgo
mMJthmtnmnnywmmerday. Inverse: Whenamithnotamnnydqdeomtgototlnbmch.
Guam: lflaleapuntilnoonﬁhenlstqeduphm. Connapoalth‘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 1H? 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: lacxecuudmo 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 cumting 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.) [fathalamus
athed.whomldlhovetheharbet? [fthehorbenhovodhimoelfﬁhenhewuldbcviolntingthemlethat
hashmoulythooepeoplowhodonotuhmthemselm Ontheothcrhonddfhodomnotshmhlmnell.
theotheruleaoysthathommtshmhlmmlf. Neitherlnpomlbleaothmoconhenosudnbubet. a) Iftheexplomr (om,sothatourpmooum wlllnotnetconhnedhemtlnconnlhalswlllhenmlc)
encoununatmthtellar.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—owq.
d) The hypothes’u is a conjunction: (13A 1) ¢ q. Wewritctlmsymholically: u—au,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. ‘Loo 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 antiBod 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. “Smithlndlonesueinnooent (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.dmehnumynthatbedidnotknowCoopeiandmom~ 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
aaidthathedidnmkmeooperwheninfactlewaawithhim. 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: KH.R—oaV.~RoV.A—oR,VbK,KoV.H—oA,H—oK.
Notethatwewereabletoconvertalloithesestatementaintocomlitionalstatementa.lnwhatlollcwswewili
sometimes mobmeddle 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 truthteller, then Carlos did It; but this
means that John is telling the truth. a contradiction. if John is the solo truthteller. then Diana must be
lying. so she did it. but then Carlos is telling the truth. a contradiction. If Carlos is the sole truthteller, then
Diana did it, but that mains John truthful. again a contradiction. So the only posibility is that Diana is the
sole truthteller. 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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 Winter '09
 Meenakshisundaram

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