old-midterm1-solutions

old-midterm1-solutions - CS 173, Spring 2010 Midterm 1...

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Unformatted text preview: CS 173, Spring 2010 Midterm 1 Solutions Problem 1: Short answer (12 points) State whether each of the following claims is TRUE or FALSE . Justification/work is not re- quired, but may increase partial credit if your short answer is wrong. (a) For all integers p and q , if p | q then q must be positive. Solution: False. (b) For all prime numbers p , there are exactly two natural numbers q such that q | p . Solution: True. (c) 8 11 (mod 3) Solution: True. (d) There is a set A such the cardinality of P ( A ) is less than two. Solution: True. (e) For all positive integers p and q , gcd( p,q ) lcm( p,q ). Solution: True. (f) For all sets A and B , P ( A B ) P ( A B ). Solution: True. Problem 2: Set theory calculation (10 points) Suppose that A = { 4 , 5 , 6 } and B = { 2 , 7 , 8 , 11 , 13 } . Calculate the values of the following expressions (recall that P ( X ) is the power set of X ). Explicitly list the contents of non-empty sets. (a) A { 4 , { 5 , 6 } , 11 } = Solution: { 4 , 5 , 6 , 11 , { 5 , 6 }} (b) Cardinality of P ( A B ) = Solution: 2 3 5 = 2 15 (c) P ( A ) P ( B ) = Solution: {} (d) { p B | p 2 A } = Solution: { 2 } (e) A { a, } = Solution: { (4 ,a ) , (5 ,a ) , (6 ,a ) , (4 , ) , (5 , ) , (6 , ) } Problem 3: Longer answers (8 points) (a) (3 points) Trace the execution of the Euclidean algorithm as it computes gcd(1012...
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old-midterm1-solutions - CS 173, Spring 2010 Midterm 1...

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