113 HW 1_Math 113 - Mathematics 113 Homework 1 Curtis...

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Mathematics 113: Homework # 1 Curtis Tekell Dr. Poirier September 8, 2010 Exercise 1. Determine whether the following functions f are well defined: (a) f : Q Z defined by a b 7→ a . (b) f : Q Q defined by a b 7→ a 2 b 2 . Solutions. (a) No. 1 = f ( 1 2 ) = f ( 2 4 ) = 2, hopefully not true. (b) Yes, this is well defined. We do not have a problem with choosing arbitrary representatives from the equivalence classes a b . For instance, assume that a b = c d . Then ad = bc , and a 2 b 2 = f ( a b ) = f ( c d ) = c 2 d 2 . Therefore a 2 d 2 = c 2 b 2 , which holds if ad = bc . Exercise 2. Let A be a nonempty set. (a) If determines an equivalence relation on A , A/ partitions A . (b) If D = { A i | i I } is a partition of A with no A i = , then there exists an equivalence relation on A such that A/ = D . Proof. (a) First, as is an equivalence relation, it is reflexive, thus a a for all a A , so each a is in some equivalence class, and S D A/ D = A , it remains to show that no nonequal B, C A/ intersect. Assume that a B C . Clearly a c and a b for all b B and c C , thus by transitivity, b c for all b B and c C , and hence C = B . It follows that two equivalence classes are either
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