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Math 109 M2 Ans

Math 109 M2 Ans - Prof Allen Knutson's Math 109 Midterm#2...

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Prof. Allen Knutson’s Math 109 Midterm #2, Nov 13, 2006 Name: SID: If you have questions, come and ask me! I really really mean it!!! 1. For each of the relations listed below, write R , S , T if the relation is r eflexive ( x, x x ), s ymmetric ( x, y, x y = y x ), or t ransitive ( x, y, z, x y & y z = x z ). You aren’t required to prove that the relations have these properties. For each such property that a relation does not have, give an example showing that the relation lacks the property. (That is to say, you are required to prove that they don’t have those properties.) Part A is done as an example: A. Let X = R , and write a b if a b . R, T. This is not symmetric because 55 89 but 89 6 55 . B. Let X = R , and write a b if b - a is an integer multiple of . R,S,T. (Some people seemed to think that 0 is not an integer multiple of , but it is.) C. Let X = R , and write a b if a is a real-number multiple of b , i.e. x R s.t. a = xb .
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