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# quiz10ANS - Math 240 Quiz 10 Name Circle One T 12:05 T 2:25...

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Math 240, Quiz 10 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) Let R be the relation on the set of ordered pairs of positive integers such that (( a, b ) , ( c, d )) R if and only if ad = bc . Show that R is an equivalence relation. Cleary, ab = ab so that ( a, b ) R ( a, b ). Also, if ad = bc then cb = da , so that if ( a, b ) R ( c, d ) then ( c, d ) R ( a, b ). Transitivity is the hardest: if ( a, b ) R ( c, d ) and ( c, d ) R ( x, y ), we need to show that ( a, b ) R ( x, y ), that is, we need to show ay = xb . We know that ad = bc and cy = dx . Multiplying the left sides and the right sides, we get that adcy = bcdx . Since dc is present in both sides, we can remove it to get that ay = bx . 2) Answer these questions for the poset ( { 2 , 4 , 6 , 9
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