Module 1-5 notes handwritten simple ann short, Study notes of Discrete Structures and Graph Theory

Download Module 1-5 notes handwritten simple ann short and more Study notes Discrete Structures and Graph Theory in PDF only on Docsity! Aiddhaarth -¥ 861694 1993 venom’ []atmemarricat | ogic A statement or proposition is a declarative statement that is either true or false (~ not both). ¢ Atomic statement - Simple / Si bine statement <Compourd datament~ Grosshtint of ata dato! with — conneckyes. ~p : Not p pa: if ptheng pq: p And q4 | p+>q :if and only if p then 9 pug: porg L> Construct the truth table: Cp—>4q)<> (~pva) > | 4 | wp [pry [p> 4 |@--g<> CrP v4) T T F T T T T T A proposition is a tautology when au the resulls ave True. Any statement with all values as False is called a6 0 ‘conbradictn. A proposition ‘that does'nt satisfy either of the two conditions given above , is called as —contingeney ._ Le Construct a truth table for the following: I- pA CpY4) 2 (peg) p 3. Cpv~4) 4 Tp 14 [~4 | pv~a | Cpy~ay— « T Foi Tr T T F T T T F Convert the fotowing to Ponr and fenF ~(~p v~q)— (p <q) =s~ C~pv~q) Vv (p<>~q) =(pAq) Vv Cperrq) : Cp A q) v[CpA4q) vCwpa~qd] =(p A 4) vtpA~aq) v (wpAq) ——+» PONF Form PCNF = w~C Remaining min terms) «=~ (Cap A~q) z pP%¥4 —————— PenF form TAREE ee Tr] Tt] Fe F F T T T T T F F T T F T F F Obtain ppwF /penrF by truth table method : (pvq) A Cr veep) A (4 vor) = [¢pvq) v¢rAor)] A [trv~p) vq Awq)| A [4 ver) vCp Awp))] =[(pveve) A (py Vor] Al cup VqvrvA Cwpvwqvr)] AGpy q Yor) ——> PCNF pone = v( Remaining max terms) ow [ow pvuq very (p¥ag vr) A (pvvg vor)] = (pAqdr) v(~pAqdwr)vCup Agar) ——> pon form Duality : The dual of a proposition that contain only the logitad operators V,A end ~ is the proposition obtained replacing each Vby A, eachA by A ,each F to T and each T+ F. The duality 4s represented as A* for a statement A f Any bet of conneckives in which every formula can be expressed as another eq: formula containing connechives from this sek is called fanctipnal Comoldte Set o- connectives deample [vse nd fA,~} are functionally complete fod ot [fe An argument is a sequence of proposition H,,H,---- Hy Called premises 5 /nypotheses followed by a proposition C , called as Conclusion . [RAIA % pins le if 6 whenever HH, ,H,----,Hy is true | else it is invalid Rule P: remcse be introduced at tat in the derivation md “a Rule T : A formule 5 may be introduced in a derivation if S is tautologically implied by one or sore of the preceding formulae in the derivation. rE pAgq >p- implification pAq >4 J 2 p> pvq - Addition q > pA4q "prog > pAq . p> pP—4 > q- Modus Ponens ~P, pVq >4q ~ Disjunctive Syllogisa “94, pq e~p - Modus Tollens P->9, QR > P->R- Hypothetical Syltogism Pv¥Q@, PR, Q7>R > R—- Dilemma PHD MH Dd vo Show that RACPVQ) is a valid conclusion from the premises PYQ , QR, PN, om. Derivation Rule P—nm P om P ~p TCt,2) —> Modus toilens Pvq Pp Tl3,4)— disjunctive sylloqesm . Ga P J J - R T(5,6) —> Modus ponens RACPVQ@) T(4, #9