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# homework1 - Math 113 Section 5 Fall 2010 Homework#1 Due...

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Math 113 Section 5 Homework #1 Fall 2010 Due: Wednesday, September 8 1. Section 0.1, Exercise 5: Determine whether the following functions f are well defined: (a) f : Q Z defined by f ( a b ) = a . (b) f : Q Q defined by f ( a b ) = a 2 b 2 . 2. Prove Proposition 2, section 0.1: Let A be a nonempty set. (a) Prove that if determines an equivalence relation on A then the set of equivalence classes A/ of forms a partition of A . (b) Prove that if { A i | i I } is a partition of A then there is an equivalence relation on A whose equivalence classes are precisely the sets A i , i I . 3. Let Z > 0 = { 1 , 2 , . . . } be the set of positive integers (sometimes called the natural numbers and denoted N ). Strange but true: there is a bijection of sets ϕ : Q Z > 0 . Determine ϕ . Hint: you may describe ϕ as a list of ordered pairs or as a formula or as any kind of diagram you like. The definition need not be unique so your answer may differ from your friend’s. 4. (a) Use the Euclidean algorithm to determine the greatest common divisor of 12 , 345 and 9876 . (b) Find a and b so that 12 , 345 a + 9 , 876 b = (12 , 345 , 9 , 876) . 5. Section 0.2, Exercise 7: If p is prime, prove that there do not exist

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homework1 - Math 113 Section 5 Fall 2010 Homework#1 Due...

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