1. Logic.pdf

Let us Recoll o The converse, inverse ond contropositive of the implicotiorl p+ q ore: Converse : Q)P Inverse : -P+-Q Contropositive : - q-+- P o Quontifiers qnd quontified Ststements : Look ot t...

Let us Recoll o The converse, inverse ond contropositive of the implicotiorl p+ q ore: Converse : Q)P Inverse : -P+-Q Contropositive : - q-+- P o Quontifiers qnd quontified Ststements : Look ot the following stotements: p : "There exists on even prime number in the set of nqturol numbers"' q : "A11 noturol numbers ore positive"' Eoch of them osserts o condition for some or oll objects in o collection..'there exists" qnd "for oll" ore colled quontifiers. "There exists" Words qll" is is colled existentiol quontifier ond is denoted by symbol :. "For colled universol quontifier ond is denoted by V ' Stqtements involving quontifiers ore colled quontified ststements. Every quontified stotement p the corresponds to o collection ond o condition' In stotement collection is 'the set of noturol numbers' ond the condition is 'being even prime'. Whot is the condition in the stotement q ? A stqtement quontified by universol quontifier V is true if o11 objects in the collection sotisfy the condition. And it is folse if ot leost one object in the collection does not sotisfy the condition. A stotement quontified by existentiol quontifier : is true if ot leost one object in the collection sotisfy the condition. And it is folse if no object in the collection sotisfy the condition' Idempotent Low p^p=p' pvp=p Commutqtive Low pYq=clvp pna-_!_!_L Associotive Low p (q n r) = (P " q) A r :- P A q A r p ^Y (q v r) = (P " q) , !:J_, q u: Distributive Low pn(q"r)=(Pnq)v(P!r) p "G;i = (p v q) n (P:) De Morgon's Low * n q) : - p Y -Q, -@ v q) = -P n -q Identity Low WF,pv F=p,pv T=T Complement Low p,.-p=F,Pv-P=T  Absorption Low pvQtxq)=p,pn@v Conditionol Low Biconditionol Low p+>q= Ex' (1) Write the converse, inverse, contropositive ond the negotion of the implicotion: "If two sides of q triqngle ore congruent then it's two ongles qre congruent.,, Solution : converse : If two ongles of o triongle ore congruent then it,s two sides qre congruent. Inverse : If two sides of o triqngle ore not congruent then it,s , two ongles ore not congruent. contropositive : If two ongles of o triongle qre not congruent then it's two sides qre not congruent. Negotion : Two sides of q triongle qre congruent but it,s two ongles qre not congruent. Ex. (2) write (q) truth volues qnd (b) negotions of the folrowing stotements : i) VxeR,x2 is positive. ii) 1xeR,x2 is not positive. iii) Every squore is o rectongre. iv) some poroilelogroms qre rectongles. Solution : o) Truth volues i) folse becouse the squore of 0 is not positive ii) true becquse the squore of 0 is not positive iii) true iv) true b) Negotions i) 1xeR,x2 is not positive. ii) yxe R,x2 is positive. iii) There exists q squore which is not o rectongle. iv) No porollelogrom is o rectongle. Ex. (3) without using truth tqble prove thot {[(r" q)n- pf--t]=t-+ t Solution : L.H.S. = {[(, , q)^ - p)-- q]  =[-(p,,q)v pf, - q (....P.9...,n.q61.d]...!l*.... ) =[(- pn - q), pfu - q (.. pg....fl.e/.$ qd:...1*.*.......) t-av9......) =[(-p v p)n(*qvp)]"- q (....D.U.kih.^+t*.. =[(r)n(- qu p))u -q t....C.gm.8!.q-rngot...!s4.....1 =(-qvp)v-q (....Id.+Jil.l...!*.*..........) qv pv-q 1.....4s.:.9..gi.s,f1g....\s.l.+.....1 = -qvp t..Id.*mfg..h*S...taur........) =q-+ p L.H.S.=R.H.S. Ex. (4) Using truth toble Solution : I u m ry V VI VII VIII IX p q -p -q peq -(peq) p^-q q^ -p (pn-q)v(qn-p) T T tr tr T F F F e. T T F- T r tr F F T F- T-T- F tr r ? T r I F F T T tr F tr F From column (VI) ond (IX) we conclude thot Ex. (5 ) Is - (p * q) equivolent to (- p)+> q ? Justify. n : J... tr J[E v \rI f L t".rp ,u? e 1* PeL 1l!*D -r r r F ? ,r F e T t T F T I T T f T F F T r- T 12.', - Ftpq (olunn r\o lV qnA Vl w....1... "t"'i Ld,clql