aa_hw_1 - Homework 1 September 8, 2010 Introduction to...

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Introduction to Abstract Algebra Homework 1 September 8, 2010 1 Section 0.1, Excercise 5: Determine whether the following functions are well defined: (a) f : Q Z defined by f ( a b ) = a . Our criteria for a well defined function is, that all representatives of the same equivalence class map to elements in the same equivalence class. For example, if a and a Í are two elements in the same equivalence class, then f ( a ) and f ( a Í ) must be in the same equivalence class for the function to be well defined. This is not the case for this function. For example, 2 2 and 1 1 are in the same equivalence class, but map to elements in different equivalence classes. (b) f : Q Q defined by f ( a b ) = a 2 b 2 . This function is well defined, since all numbers in the same equivalence class map to elements in the same equivalence class, e.g. 3/4 and 6/8 map to 9/16 and 36/64 respectively. These are equivalent, since 36/64 = 9/16. 2 Prove Proposition 2, section 0.1: Let A be a nonempty set. We wish to prove that (a) Theorem 1. If determines an equivalence relation on A , then the set of equivalence classes of A/ of forms a partition of A . Proof. The proof consists of two parts. We first prove, that any two equivalence classes are distinct. Second we prove, that the union of all equivalence classes make up all of A . (a) Let a,b A . Assume that [ a ] [ b ] Ó = and [ a ] Ó = [ b ], and b Ó∈ [ a ] and a Ó∈ [ b ]. Now suppose there exists some x [ a ] and x [ b ]. Then x a and x b . Due to symmetry, a x . Now, from the property of transitivity, a b , which means a [ b ], so we have a contradiction, and the sets must be equal or distinct. (b) For all a A : a [ a ], so clearly the union of all such [ a ] cover all of A . Formally put, Û i I A i = A. (b) Theorem 2. 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 . Proof.
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aa_hw_1 - Homework 1 September 8, 2010 Introduction to...

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