42_2_sol1 - y 3 = (2 k + 1) 3 = 8 k 3 + 12 k 2 + 6 k + 1 =...

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San Jose State University Department of Mathematics, College of Science Fall 2009 MATH 42, Discrete Mathematics, section 2 SOLUTION 1 1. (a) False (b) The inverse is : if there is not a hurricane then the beach does not erode. (c) ( p r ) ( q r ) and ( p q ) r are logically equivalent because their corre- sponding columns in the truth table below are identical. p q r p r q r ( p r) ( q r) p q ( p q) r T T T T T T T T T T F F F F T F T F T T T T T T T F F F T F T F F T T T T T T T F T F T F F T F F F T T T T F T F F F T T T F T 2. (a) i. ¬∃ yL ( Jill, y ) ii. x yL ( x, y ) (b) Take x = 1 and y = 0. (c) i. True ii. False 3. (a) (1) ¬ r → ¬ q (premise) (2) ¬ r (premise) (3) ¬ q (modus ponens and (1), (2)) (4) p q (premise) (5) p (disjunctive syllogism and (3) (4)) (b) type of proof: proof by contradiction Proof: Assume that y is not even. Hence y is odd i.e. y = 2 k + 1 for some integer k . Now
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Unformatted text preview: y 3 = (2 k + 1) 3 = 8 k 3 + 12 k 2 + 6 k + 1 = 2(4 k 3 + 6 k 2 + 3 k ) + 1, which is odd. It contradicts that y 3 is given as even. 1 4. (a) 2 (b) , S, { 2 } , { b } (c) 27 (d) 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 (e) At least 8 and at most 15 people have visited both Russia and China. 5. (a) domain = all positive integers range = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } (b) Yes, f is onto because given any y Z , take n = 3 y Z then f ( n ) = f (3 y ) = b 3 y 3 c = b y c = y . (c) an example is g : N- N dened by g ( x ) = x + 1 6. (a) a 1 = 1 , a 2 = 2 , a 3 = 3 , a 4 = 3 , a 5 = 4 (b) 14 (c) a n = 3 n + 2 2...
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42_2_sol1 - y 3 = (2 k + 1) 3 = 8 k 3 + 12 k 2 + 6 k + 1 =...

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