e210k - MAD 2104 Quiz II and Key May 13, 2010 Prof. S....

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MAD 2104 May 13, 2010 Quiz II and Key Prof. S. Hudson 1) [45 pts] Find a counterexample to each of these, where the domain is Z , the set of integers: a) x, y, ( x 2 = y 2 x 3 = y 3 ) b) x, y, ( y 2 = x ) c) x, y, ( y 2 = x 3 ) 2) [30pts] Let A i = { i,i +1 ,i +2 ,... } for every i Z . Find these sets (we did one of these in class, but I changed the other slightly): a) S i =1 A i b) T 5 i =1 A i 3) [25 pts] Prove or disprove that the product of a nonzero rational number and an irrational number is irrational. If you need to use the back, leave a note here. Remarks and Answers: The average was about 57 / 100. The unofficial scale for this quiz is A’s = 70 to 79 B’s = 60 to 69 C’s = 50 to 59 D’s = 40 to 49 1a) Let x = 1 and y = - 1. Then x 2 = y 2 , but x 3 6 = y 3 . Notice that the counterexample includes values for both x and y because both are quantified by . 1b) Let
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e210k - MAD 2104 Quiz II and Key May 13, 2010 Prof. S....

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