quiz9ANS - M R = 1 1 1 1 1 1 ±ind R-1 and R 2 M R-1 = 1 1...

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Math 240, Quiz 8 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1) Determine whether the relation R on the set of all integers is reFecive, symmetric, antisymmetric, and/or transitive, where ( x, y ) R xy 1. It is not reFexive ((0 , 0) / R ). It is symmetric, since multiplication is commu- tative. It is not antisymmetric ((1 , 2) and (2 , 1) are in R , but 1 n = 2). ±inally, it is transitive: if xy 1 and yz 1, then y n = 0, x n = 0 and z n = 0. So x and y have to have the same sign, as do y and z . Thus, x and z have the same sign, so xz must be positive, so xz 1. 2) Let R be the relation represented by the matrix
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Unformatted text preview: M R = 1 1 1 1 1 1 ±ind R-1 and R 2 M R-1 = 1 1 1 1 1 1 , while M R = 1 1 1 1 1 1 1 1 1 3) Let R be the relation { ( a, b ) | a n = b } on the set of integers. What is the reFexive closure of R ? Since we are simply adding in the set of all { ( a, a ) | a ∈ Z } = { ( a, b ) | a = b } , we get that the reFexive closure is the set of pairs of integers, or Z × Z . Bonus (1 pt): How many states does Wisconsin border? ±our: Michigan, Minnesota, Iowa, and Illinois. 1...
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This note was uploaded on 08/11/2008 for the course MATH 240 taught by Professor Miller during the Fall '08 term at University of Wisconsin.

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