Name printed Student ID Section or TA s name and time CMSC 250 Quiz 6 ANSWERS Wednesday Mar 3 2004 Write all answers legibly in the space provided The number of points possible for each question is indicated in square brackets the total number of points on the quiz is 30 and you will have exactly 20 minutes to complete this quiz You may not use calculators textbooks or any other aids during this quiz 1 24 pnts Disprove by counter example or Prove each of the following a The product of any rational number with an integer is a rational number x Q y Z xy Q PROOF Let x by arbitrary in Q and let y be arbitrary in Z Since x is rational a b Z x ab where b 6 0 by definition of rational The product of x and y can be written as y ab by substitution After multiplying xy ya b Since ya Z by closure of Z in multiplication and b Z and b 6 0 because it was defined as such above Therefore xy Q by definition of rational x Q y Z xy Q by generalizing from the generic particula b For all integers n if n is odd then n2 is odd n Z n Z odd n2 Z odd PROOF Let n be arbitrary in Z Assume odd n Z Since n is an odd integer a Z n 2a 1 by the definition of odd Since n 2a 1 n2 2a 1 2 by squaring both sides 2a 1 2 4a2 4a 1 2 2a2 2a 1 by algebra Since 2a2 2a is an integer by closure of integers during addition and multiplication n2 is also even by the definition of odd n Z odd n2 Z odd by closing the conditional world n Z n Z odd n2 Z odd by Generalizing from the Generic Particular c For all integers n and m n m m False n 2 and m 5 n m 2 5 3 which is not greater than 5 2 6 pnts State Yes or No for each of the following only a small justification is needed for your answer not a complete proof Assume a b and c are integers and x y and z are rationals for all of the following questions a NO If a b and c are even a b c 2 is also even justification a 2 b 2 and c 2 then a b c 6 and 6 2 is 3 which is odd b NO x y x y justification x 2 and y 2 x y 0 and xy 4 but 0 6 4 c NO If a b and x y then a x b y justification a 1 and b 2 and x 4 and y 5 while it is true that 1 2 and 4 5 it is not true that 4 10
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