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# e210k - MAD 2104 Quiz II and Key Prof S Hudson 1[45 pts...

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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 x = - 1. There is no y R such that y 2 < 0, so y 2 = x is always false. Notice that the counterexample includes a value only for x , because only x is quantified by .

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e210k - MAD 2104 Quiz II and Key Prof S Hudson 1[45 pts...

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