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e411k - MAD 2104 Quiz 4 Key Prof S Hudson 1[30pts Show that...

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MAD 2104 Oct 13, 2011 Quiz 4 Key Prof. S. Hudson 1) [30pts] Show that the union of two countable sets is always countable. As usual, a large part of your grade will be based on your explanation (use enough words!). 2) [20pts] How many bit strings are there of length 5 or less ? 3) [20pts] Compute the Boolean product A B , where A = 1 0 1 1 0 0 and B = 1 1 0 0 1 1 4) [30pt] Choose ONE proof. Use induction. You may answer on the back. a) Prove for every positive integer n , that 1 2 + 2 2 + · · · + n 2 = n ( n + 1)(2 n + 1) / 6. b) Prove for every positive integer n , that 3 divides n 3 - n . Remarks and Answers: The average of the top 20 grades was approx 74 / 100. The lowest grades were mostly on Problem 1, but the rest were OK on average. The scale is A’s 82 - 100 B’s 72 - 81 C’s 62 - 71 D’s 52 - 61 As on Quiz 3, I have estimated your semester grade in the upper right corner. But this time, I added your best 3 out of 4 quiz grades, so far. The average for this was about 240 / 300, which is (of course) higher than before. I used a scale of A’s 270-300, B’s 240-270, etc. Because of the change in the calculations (eg now dropping the lowest grade), the new
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