Discrete Mathematics with Graph Theory (3rd Edition) 31

# Discrete Mathematics with Graph Theory (3rd Edition) 31 -...

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Chapter 1 29 (q v (-.p)) <===} ((-.q) -t (-.p)). Hence part (a) again gives (p -t q) <===} ((-.q) -t (-.p)) , which is Property 11. 6. (a) ((p-tq)-tr) <===} (((-.p)Vq)-tr) <===} ((-.((-.p)Vq))Vr) <===} ((PA(-.q))Vr) <===} ((P V r) A ((-.q) V r)) <===} ((P V r) A (-.(q A (-.r)))). (b) [p -t (q V r)] <===} (-.p) V (q V r) <===} -.p V q V r (associativity) <===} [-.(p A (-.q)] V r (De Morgan) <===} [PA (-.q)]-t r. 7. (a) ((P V q) A r) V ((P V q) A (-.p)) <===} ((p A r) V (q A r)) V ((p A (-.p)) V (q A (-.p))) <===} ((p A r) V (q A r)) V (0 V (q A (-.p))) <===} ((p A r) V (q A r)) V (q A (-.p)) <===} (PA r A q) V (pA r A (-.q)) V (q Ar Ap) V (q A r A (-.p)) V(q A (-.p) A (-.r)) <===} (p A q A r) V (p A (-.q) A r) V ((-.p) A q A r) V(( -.p) A q A (-.r)) (b) [PV(qA(-.r))]A-.(qAr) <===} [p V (q A (-.r))] A [(-.q) V (-.r)] <===} ( [p V (q A (-.r))] A (-.q)) V ([p V (q A (-.r))] A (-.r) ) <===} (p A (-.q)) V (q A (-.r) A (-.q)) V (p A (-.r)) V (q A (-.r) A (-.r)) <===} (p A (-.q) A r) V (p A (-.q) A (-.r)) V 0 V (p A q A (-.r)) V (p A (-.q) A (-.r)) V (p A q A (-.r) V (( -.p) A q A (-.r))
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