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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 equal or disjoint, and thus A/ defines a partition on
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online