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Discrete Mathematics with Graph Theory (3rd Edition) 14

Discrete Mathematics with Graph Theory (3rd Edition) 14 -...

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12 Solutions to Exercises (e) p q r q->r p -> (q -> r) p/\q (p/\q)Vr (p -> (q -> r)) -> «p /\ q) V r) T T T T T T T T T F T T T F T T F T T T T F T T F F T T T F T T T T F F F T T T T F F T T F F F F T F F T F F F F F F T T F F F 2. (a) If p -> q is false, then necessarily p is true and q is false. (This is the only situation in which p -+ q is false.) We construct the relevant row of the truth table for (p 1\ (...,q)) V ((...,p) -+ q). I (p/\(.q)) ~ (.p)->q) I (b) [BB] There are three situations in which p -+ q is true. The question then is whether or not the truth value of (p 1\ ( ...,q)) V ((.p) -+ q) is the same in each of
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Unformatted text preview: truth table. p q .q p /\ (.q) .p (.p)->q (p/\ (.q)) V (.p) -> q) T T F F F T T F F T F T F F As shown, (p 1\ (.q)) V ((. ..,p) -+ q) has different truth values on two occasions where p -+ q is true, so it is not possible to answer the question in this case. 3. [BB] ~~~--+-~--~--~~~~~~~~-+~~~ F [p -+ (q /\ (.r))] V [r +-+ ((. ..,s) V q)] T 4. T [p -+ (q 1\ (...,r))] V [r +-+ ((. ..,s) V q)] T 5. (a) [BB] p q pl\q pVq (p 1\ q) -+ (p V q) T T T T T T F F T T F T F T T F F F F T The final column shows that (p /\ q) -+ (p V q) is true for all values of p and q, so this statement is a tautology....
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