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Unformatted text preview: CS 173, Fall 2009 Midterm 1 Solutions Problem 1: Short answer (12 points) State whether each of the following claims is true or false. Justification/work is not required, but may increase partial credit if your short answer is wrong. (a) − 6 | − 3 Solution: False. The magnitude of -6 is larger, so there’s no way you could have an integer k such that − 6 k = − 3. (b) For every integer x , if x is a negative prime number then x 2 < 0. Solution: True. It’s vacuously true because no value of x will make the hypothesis true. (c) ⌈ x ⌉ < ⌊ x + 1 ⌋ for any real number x ? Solution: False. Suppose x = 3 . 1. Then ⌈ x ⌉ = 4 = ⌊ x + 1 ⌋ . (d) For any set A , A ∈ P ( A ). Solution: True, because A is a subset of itself. (e) For any positive integers p and q , if lcm( p, q ) = pq , then p and q are relatively prime. Solution: True. lcm( p, q ) = pq gcd( p,q ) and the definition of “relatively prime” is gcd( p, q ) = 1. (f) P ( A ∪ B ) = P ( A ) ∪ P ( B ), for any sets A and B . Solution: False. A ∪ B is often larger because it can contain “mixed” subsets, with some elements from A and some elements from B . Problem 2: Calculation (11 points) Calculate the values of the following expressions. Your answer to (e) must be in closed form, i.e. not using a summation. Recall that P ( A ) is the power set of A ....
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- Spring '08
- Natural number, Prime number, Euclidean algorithm, House Bill