midterm1_sol - Faculty of Mathematics University of...

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Faculty of Mathematics University of Waterloo MATH 135 MIDTERM EXAM #1 Fall 2005 Monday 17 October 2004 19:00 – 20:15 Solutions 1. (a) Construct truth tables for the two statements [7] P OR ( Q OR R ) and ((NOT P ) AND (NOT Q )) = R Solution: For the first statement: P Q R Q OR R P OR ( Q OR R ) T T T T T T T F T T T F T T T T F F F T F T T T T F T F T T F F T T T F F F F F For the second statement: P Q R NOT P NOT Q (NOT P ) AND (NOT Q ) ((NOT P ) AND (NOT Q )) = R T T T F F F T T T F F F F T T F T F T F T T F F F T F T F T T T F F T F T F T F F T F F T T T T T F F F T T T F (We could have combined these two tables into one table if our paper was wide enough.) (b) Are the statements P OR ( Q OR R ) and ((NOT P ) AND (NOT Q )) = R equivalent? [2] Give a one sentence reason. Solution: The statements are equivalent, because the last columns in the two truth tables in part (a) are identical.
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MATH 135, Midterm #1 Solutions Page 2 of 5 2. (a) If the universe of discourse is the integers, is the statement x y, x 3 + y = 0 TRUE or [2] FALSE? Explain your answer. Solution: This statement is TRUE, because for every integer x , we can choose y = - x 3 (which is an integer), which satisfies x 3 + y = 0. (b) If m and n are integers, is the statement “If m + n is odd, then m is odd or n is odd” [2] TRUE or FALSE? Explain your answer. Solution: This statement is TRUE, because if m + n is odd, then m and n cannot both be even (otherwise m + n would be even), so either m is odd or n is odd. (c) If m and n are integers, is the statement “If m is odd or n is odd, then m + n is odd” [2] TRUE or FALSE? Explain your answer. Solution: This statement is FALSE, because if m = 1 and n = 3 (or in general if m and n are both odd), then m + n = 4, which is even, not odd.
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