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Discrete Mathematics with Graph Theory (3rd Edition) 25

# Discrete Mathematics with Graph Theory (3rd Edition) 25 -...

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Section 1.3 which we check with a truth table. p q 8 T T T T F T F T T F F T T T F T F F F T F F F F pVq p--'>8 T T T T T T F T T F T F T T F T qV8 T T T T T F T F * * * * 23 There are four rows when the premises are true and, in each case, the conclusion is also true. The argument is valid. (d) Since « -,q) " r) ¢:::=} -, (q V (-,r)) and -, (p " 8) ¢:::=} [( -,p) V (-'8)] ¢:::=} (p --'> (-'8)), the premises become (q V (-,r)) --'> p P --'> (-'8) so the chain rule gives (q V (-,r)) --'> (-'8), which is logically equivalent to (-,( q V (-,r))) V (-'8), which is the desired conclusion by a law of De Morgan. 4. (a) [BB] Since [(-,r) V (-,q)] ¢:::=} [q --'> (-,r)], the first two premises give p --'> (-,r) by the chain rule. Now -,p follows by modus tollens. (b) This argument is not valid. If q and r are false, p and 8 are true, and t takes on any truth value, then all premises are true, yet the conclusion is false. (c) [BB] This argument is valid. Since p +-t (t V 8), we can replace t V 8 with p so that the premises become p V (-,q), p --'> (p V r), (-,r) V p. The first of these is logically equivalent to q --'>
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