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

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