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Name:__________________________________ ID:______ ___________________________ ECS 20: Discrete Mathematics Midterm 1 January 25, 2006 Notes: 1) 2) 3) 4) 5) 6) quizz is open book, open notes. No computers though… You have 30 minutes, no more: I will strictly enforce this. You can answer directly on these sheets (preferred), or on loose paper. Please write your name at the top right of each page you turn in! Please, check your work! There are 6 questions total, each valued 5 points. I will grade however over a total of 25, i.e. one question can be considered “extra credit”. You choose! Part I: logic (2 questions, each 5 points; total 10 points) 1) Use either a truth table or logical equivalences to show the equivalence : p F q T p q p. 2) Show that the statement p q p q is not a tautology by giving examples of propositions p and q for which the implication is false 1 Name:__________________________________ ID:______ ___________________________ Part II: proofs (2 questions, each 5 points; total 10 points) 1) Prove that if N is an even number greater or equal to 4, then 2N-1 is not prime (Hint a number A is not prime if it can be written in the form BxC, where both B and C are integers strictly greater than 1). Justify each step. 2) The proof below has been scrambled. Please put it back in the correct order. Claim: For all n ≥ 9, if n is a perfect square, then n-1 is not prime. Since (n-1) is the product of 2 integers greater than 1, we know (n-1) is not prime (1) Since m ≥ 3, it follows that m-1 ≥ 2 and m+1 ≥ 4 (2) Let the perfect square n ≥ 9 be given (3) This means that n-1 = m2-1 = (m-1)(m+1) (4) There is an integer m ≥ 3 such that n=m2 (5) 2 Name:__________________________________ ID:______ ___________________________ Part III: Set theory (2 questions, each 5 points; total 10 points) 1) Let A and B be two sets. Show that if A B B then A B A 2) Let A and B be two sets. Show that if A B A then B B A A 3