Discrete Mathematics with Graph Theory (3rd Edition) 16

Discrete Mathematics with Graph Theory (3rd Edition) 16 -...

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14 Solutions to Exercises p q r 8 t -,t (-,t) v P 8 -+ [(-,t) v p) T T T T T F T T F F F T T F F F -,q (-,q)-+r [8 -+ «-,t) v p)) v [(-,q) -+ r) F T T T F F 9. We are given that A is false for any values of its variables. (a) [BB] An implication p - q is false only if p is true and q is false. Since A is always false, A - '13 is always true. So it is a tautology. (b) An implication p - q is false only if p is true and q is false. Since A is false and the tautology '13 is true for any values of the variables they contain, '13 - A is always false. So it is a contradiction. 10. (a) The tables below show that when all three variables p, q and r are false, p - (q - r) is true, whereas (p - q) - r is false. Thus these statements have different truth tables and hence are not
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Unformatted text preview: logically equivalent. I ; I ; I : I q; rip -+ ~ -+ r) I (b) The compound statement is false. 11. (a) [BB] p q T T T F F T F F (b) p q -,p T T F T F F F T T F F T (c) [BB] p q T T T F F T F F (d) p q pVq T T F T F T F T T F F F pVq F T T F (-,p) t\ q P v -,p) t\ q) (p v -,p) t\ q)) V q F T T F T T T T T F F F pVq pVq (p V q) -+ (p V q) F T T T T T F F p+-+q -, (p +-+ q) T F F T F T T F T T T T. The truth table shows that (p '::!. . q) -(p V q) is true for all values of p and q, so it is a tautology. Columns three and five are the same. So the truth values of p '::!. .. q and --, (p f-+ q) are the same for all values of p and q. Thus these statements are logically equivalent....
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