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# exam1sol - 1(10 pts Use the laws of logic to prove the...

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1) (10 pts) Use the laws of logic to prove the following expression is a tautology: (Note: You must list each rule you use. The only rules you can combine together in one step are the commutative and associative rules to reorder and group terms more quickly. Hint: First try to “distribute” the not signs before you simplify any other part of the expression.) ( (p r) ( (q r) p) ) ¬ ( ( ¬ (p q) ) p ) ( (p r) ( (q r) p) ) ¬¬ ( p q) ¬ p (De Morgans) ( (p r) ( (q r) p) ) ( p q) ¬ p (Double Negation) ( ¬ p p) ( (p r) (q r) (p q)) (Commutative+Associative over or) T ( (p r) (q r) (p q)) (Inverse Law) T (Domination Law) 2) (6 pts) True or False, circle the correct answer, no partial credit on this one. a) If A B, then |A| |B| TRUE B contains all elements of A, thus it must have at least as many elements as A. b) If A={1,3,5,7} and B={2,4,6,8}, then TRUE A = A – B. Since A and B are disjoint, the equality does hold. c) If A ( ¬ B) = C, then C and B are TRUE disjoint sets. We need to show that C B= . We have C ⊂¬ B, since C (A ∩¬ B). Thus, for any element x, if x C, then x ∈¬ B. But, C B is non-empty only if there exists an element x such that x C and x B. But this element does not exist because all elements in C are also in ¬ B. Thus, we must have C B= . 3) (6 pts) Establish the following argument by the rules of implication. (Note: you must state the rule you use (or premise) for each step in your proof.) p p q (q r) s -------------- s 1. p (Premise) 2. p q (Premise) 3. q

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exam1sol - 1(10 pts Use the laws of logic to prove the...

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