Discrete Mathematics with Graph Theory (3rd Edition) 30

# Discrete Mathematics with Graph Theory (3rd Edition) 30 - A...

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28 Solutions to Review Exercises (b) A truth table shows that this is neither a tautology nor a contradiction. p q p----+q pVq (p --+ q) ----+ (p V q) T T T T T T F F T T F T T T T F F T F F (c) A truth table shows this is a tautology. p q r -.q p 1\ (-.q) [p 1\ (-.q)]----+ r p V [(p 1\ (-.q)) ----+ r] T T T F F T T T F T T T T T F T T F F T T F F T T F T T T T F F F T T T F F T T F F F T F F F T T F F F T F T T (d) This is neither a tautology nor a contradiction. p q r -.q (-.q) 1\ r pVq T T T F F T T F T T T T F T T F F T F F T T T F T T F F F T T F F T F T F T F F F T F F F T F F [( -.q) 1\ r] --+ (p V q) (p V q) ----+ [(-.q) 1\ r] (p V q) +-+ [( -.q) 1\ r] T F F T T T T F F F T F T F F T F F T F F F T F 4. Assume that some set of truth values on the variables makes A true. If '13 were false, this would make
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Unformatted text preview: A ----+ '13 false and '13 ----+ A true, contradicting logical equivalence. So '13 must be true also. Similarly, if '13 is true, then A must also be true. We conclude that A is true if and only if '13 is true. This means A and '13 are logically equivalent. 5. (a) Since A <==> '13, we know that A is true precisely when '13 is true. Since '13 <==> e, '13 is true precisely when e is true. Hence A is true if and only if e is true, that is, A <==> e. (b) Property 12 says (p ----+ q) <==> «-.p) V q). Clearly, «-.p) V q) <==> (q V (-.p)) , so part (a) tells us (p ----+ q) <==> (q V (-.p)). But Property 12 also says «-.q) ----+ (-.p)) <==> (-.(-.q) V (-.p)) and clearly (-. ( -.q) V (-.p)) <==> (q V (-.p)). So we have (p ----+ q) <==> (q V (-.p)) and also...
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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