homework1solutions_Math 113

homework1solutions_Math 113 - Math 113 Section 5 Homework...

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Unformatted text preview: Math 113 Section 5 Homework #1 Solutions 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 . f is not well defined: Let a b be another representative of a b (that is, a = a and b = b but a b = ab ). Then f ( a b ) = a = a = f ( a b ) . (b) f : Q Q defined by f ( a b ) = a 2 b 2 . f is well defined: This is because the codomain is Q . Let a b be another representative of a b . Then a b = ab = ( a b ) 2 = ( ab ) 2 = a 2 b 2 = a 2 b 2 = a 1 b 2 a b = f ( a b ) = f ( a b ) . 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 . Let a A and a = { b A | b a } . We need to show (1) t a A a = A and (2) a b = if a = b . (1) Every element is in some equivalence class. In particular, a a . (2) Assume a b = . Then there exists a c a b . Therefore c a and c b . Because is an equivalence relation, a b so a b and b a . Further, for any c a , c b and vice versa. Thereforeand vice versa....
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homework1solutions_Math 113 - Math 113 Section 5 Homework...

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