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# Unit 3 MATH 1301-006

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This 23 page Class Notes was uploaded by Jaquelina Rodriguez on Sunday September 18, 2016. The Class Notes belongs to MATH 1301-006 at University of Texas at Arlington taught by Sonja Godeken in Fall 2016. Since its upload, it has received 5 views. For similar materials see Contemporary Mathematics in Math at University of Texas at Arlington.

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Date Created: 09/18/16

$1lotl,k Math 1301 Logic: Simple Statements and Negations Obiectives lnterpreand expresssimple statementin symbolicform. Expressnegationsof statements. lnterpreand expresscompound statementsin symbolicorm using connectors. Statement: has a truthvalue Texas isthe largeststate.(p) F (q) Oklahoma borders Texas. 1 Negation: (-) state.(^p 1,.Texas is nr*the largest )T 2. Oklahoma c\urrr\ border Texas.(^ .1)tr Watch quantifiers,(all,none, some) especiallywhen negating. All fishcan fly.(f) .\s\^ Lov"rnu\ t\v\ (-f) ? Sovn( Some fishcan fly.s) (-s) ? No tttvr Cu\n t\\4 No fishcan fly.(n) tc,t.t\$ (-n) ? Sunre Some fish cannotfly.(c) (-c) ? n\\ $tttn Ltrvl t\9 Allare .1. None are Some are Some are not Coniunction: (A) and, but, howeveq nevertheless Texas isthe largest stateand Oklahoma bordersTexas.( Pnq ) Texas isn'tthe largest statebut Oklahoma borders Texas. ( - p^ q ) Disjunction: (v) or Let s= ltissnowing. Let r The roads are icy. = 1,.ltissnowing and the roads are icy. S A R 2. ltis not snowing and the roads are not icysn- 3. Either the roads are icy oritis snowing. rvs 4. ltis not true that,itissnowing and the.^.(SAr roads are icy. 5. ltisn't snowing but the roads are icy.n" SnY< Two final connectors: If"..then {conditlqqal} -) Antecedent )Consequent l{ -thtn_ Ifitsnows today then the roads are icy. S "\F S th{nY. " -1'- Itsnows today if and onlv ifthe roads are icy. $a-1 'r- FormalName Symbol Read $ymbolic Form t*Not" *l) lltr{re! Conjunction n "And'n phq I Disjunction V "0r"' pvq ; Conditional ---+ "lt"-then" p"'+tt I Biconditional <.-r "lfandonlyf" P#q Lets=Youstudy. Let g You getgood grades. = Letc=Youearnacredit. 1,.lfyou study thenyou get good grades. S -)3 2.You get good grades ifand onlyifyou study. q<-) s 3. ltisnot thecase that,ifyou studyyou do not getgood grades' n- cs l ^- g ) - 4. lfyou studyor you getgood grades,then you earn acredit. tsvq) C Let: d=dinner atAntonio's s= includes soup a= includes salad o= includes potatoes p=includes pasta c= includescarrots e= includes peas c[ -: CCSvcr) n Covp)Atcve )) Summary Can you writesimple statements in symbolic form? u' Can you expressnegations of statements? *Do you feel comfortablewriting compound sentences, using the connectors and, s1;conditional and biconditional? Math 1301 Truth Values and Truth Tables for *, n, V Obiectives 'Constructruthtableand determinetruth valuesfonegationsconjunctionand disunctins Truth Table forNegation p- Sallyplays the piano. \:\utr vn<.n\ =1 L Co:e) q- Roberto willsing. Truth Table for Conjunction 3 sta1.*mr:)q tcL\) € xc^."nPl-" T T S0r\\.1r\\ t)\x".4 P\unu On cA T F ho \ocrf0 s^tt\\ Srvrt4 F T F F CO(] n Lt uvy.r Clv.0\ Cn\q ttm-e O 3r.J " o\rc" \\ trve \\ w\rtn hxrtvr Purrt- trut. Truth Table forDisjunction P = Sallyplays the piano. or Q = Roberto willsing. * $"tq t\vv1,.n- U d\l1uv'rLttt,o T T \\0t^'t \\ tu\\( \) T F hutvr 0K W141fl F T tL\\c F F Determine truthvaluesfor pV *Q T T T F F T F F How many casespossible? Ast nl.*rnn=) Q cq)&l nZ What ifthere were3 statements tt&t? Aza6 TT-1 r FT T T t\\- F FT T ( TT T r fr -T T TFT F Ff € F € T.T a TT T T rT F T r1 T F f f T .I TF T T F rr l tr F \- {- ;\ 5 hours"s S4 orrver.ueorFalse? Iisfalseha30% ofAmerican,*.O hoursosleepeach night@%get5 hoursosleepdach night. 4-c FF-rn (v Summary .Construct truth tables forcompound statements made of 1,2, dnd 3- statements? 'Find the truth value ofa singlecase without atable? Math 1301 ,l ,1 Truth Tables, If-then and Iff Obiectives *Construtruttablesddetermineruth valueoftatementusingondtiols andbiconditionals. ConditionI(lf-then) p Youbuy dinner. -T = you TT Q =|take to TF F themovies. FT t+\ou buS C\nner T thrn \ ttrrttAU FF T \o rns{gi When is conditionaflse? hrhtvr C^Jrw"\t"-lto\r(" TT T F Tir r TF F T -T,l T FT T T F]T F T I FF F F i"i T x Biconditiol--lff p= Youbuy dinner. .T TT TF Q= |takeyouto F the movies. FT ; .T FF When isabiconditionalue? \4.)vr hotvr:Yo\YeLq-nrr hmw tht \q\^^rtY\!\\VU\\)( TT *i- t I TF .T r { F, T T F r -I FF T T T T T T T F T F T T F F F T T F T F F F T F F F Consider: IfSpike Lee is amovie director orHalle Berry is aschool teacher; then George Clooney isa circusclown. d = Spike Lee is amovie director. t= Halle Berry is aschool teacher. c =George Clooney isa circus clown. T T T T T F T F T T F F F T T F T F F F T F F F Contradiction-answer column allF's Tautology-answer column allT's lmplication-all T'sfor aconditional CredNeedfor aA.A. Ilcgree T or F? ltisfalsethat if9 creditsare needed fora Communicafions degree 'issues. 6 or 4crediti are needed for a Humani-tjes degree, then Humani6ies' credits 5ie needed for a triathematMath degfee. C cI'[)*:t TT n- t -t-l F Math 1301 Equivalent Statements Obiectives . Determinthelogicaequivalenof statements. p< )q isequivalentto ) q)n (q ) p) /o (p< <=+[(p + q)A(q + p)l >q) Same truthvaluesin answer columns. T T T T T rT: T F F T F I F T F F T F F F T F F r ls ^,(p q) equivalento -p V-cN ? :r .: T F F F tr T F -T T r { T ,F- T T r T T f F F ,I F T T What aboutthe negatioofadisjunction? *r TT F r r F TF ( -r f r T -T FT € T f f FF T tr T T T DMorgaLaws t-(pAq)er*pVq Z.(pvq)*+pA*q filCcoo\.C rs olut1= 1.My A/C doesnotcoolandit isn'tet. a cA*Q {L ccvl) Notethat - (-p) -p h^d 2.The roasishotbutiisnot weldone. *r) hn L(*h) ^cl ^ dttJrlnctwv^r) 1-(,n'h V d) TT T TF-T F.T f FF? ,- TTF T T { tr TFT tr -rT T F.T { F r FFT l- F T F F (p)q)<=(-p V q) and -( p)q)<=+(pn-q) dcrtlY:r.ltr co-r: t1 s<tlrot<J.- 1. Ja nette does not bdy a rieww car or she sells'9d (Lv^ her old car. V C n-) o \".J 1. It isfalse that if Lam=told then the heater is WorkinS. 11, Ca-€'u-i) n '{^,-'UU C Variations of Conditional Statement Name Symbolicorm Read Conditional p'-+ q "lf,thenqo' Conversoftheconditional qlp '"lfthenp" *P--'+*cl 4. Inversofthconditional "Inotp,thenot{" Contrapositivthconditional -q-*p "lnotq,then not Write the converse, inverse and contrapositive for: lfIgo to Ita[ythen Ibuy a leather bag. 'L \2 C: b *--? i l: n, \.-:n- CP: \- \a '*) 4= | True or false? lfa natural number isnot divisibleby 7 then a natural number is not divisibleby 14. Contra positive: \tt o\nclt}b \5chutstbu h9 \q htn r$ C\ruttrhtl h\,,5-t q(5.T{LUZ

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