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240fall06exam1sol - MATH 240 EXAM 1 100 points(R.A.Brualdi...

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MATH 240; EXAM # 1, 100 points, October 17, 2005 (R.A.Brualdi) TOTAL SCORE (10 problems; 100 points possible): Name: These R. Solutions TA: Luanlei Zhao (circle time) Mon 9:55 — Mon 11:00 — Wed 9:55 — Wed. 11:00 1. [10 points] Using the method of proof by contradiction , give a proof of the following proposition: If n is a positive integer, then not both n and n + 2 are perfect squares. Suppose n is a positive integer, and that both n and n + 2 are perfect squares. Then n = k 2 and n + 2 = l 2 where l > k . Since n + 2 is 2 more than n , we have k 2 + 2 = l 2 and so 2 = l 2 - k 2 = ( l + k )( l - k ) . Since 2 is a prime, l + k = 2 and l - k = 1. Adding these two equations we get 2 l = 3 or l = 3 / 2, contradicting that l is an in integer. This contradiction shows that not both n and n + 2 are perfect squares. 2. [8 points] Let P ( x ) and Q ( x ) be predicates where the universe of discourse for x is some set U . Let A = { x : P ( x ) is true } and let B = { x : Q ( x ) is true } be the truth sets of P ( x ) and Q ( x ), respectively. Circle all the predicates below that have truth set equal to A B ?
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