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10Ma5aHw1Soln

# 10Ma5aHw1Soln - M a 5a HW ASSIGNMENT 1 FALL 09 SOLUTIONS...

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Ma 5 a HW ASSIGNMENT 1 FALL 09 SOLUTIONS The exercises are taken from the text, Abstract Algebra (third edition) by Dum- mitt and Foote. We have followed the book’s notation here for Z + as the set of all positive integers. (This is not standard, and many use N to denote this set, reserving Z + for the set of non-negative integers.) Page 4, 7 . Proof of Reflexivity: a a ⇐⇒ f ( a ) = f ( a ). Proof of Symmetry: a b ⇐⇒ f ( a ) = f ( b ) ⇐⇒ f ( b ) = f ( a ) ⇐⇒ b a . Proof of Transitivity: a b and b c ⇐⇒ f ( a ) = f ( b ) and f ( b ) = f ( c ) = f ( a ) = f ( c ) ⇐⇒ a c . These all follow from the fact that = is an equivalence relation. The equivalence class for a A is the set { x A : x a } ⇐⇒ { x A : f ( x ) = f ( a ) } , which is the fiber of f over f ( a ). Therefore the equivalence classes are the fibers of f . Page 8, 6 . First some philosophy. If a logical statement A = B needs to be proved, then it is enough to prove a statement equivalent to it, for example its contra-positive statement ̸ B = ⇒ ̸ A , where ̸ B (resp. ̸ A

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10Ma5aHw1Soln - M a 5a HW ASSIGNMENT 1 FALL 09 SOLUTIONS...

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