hwk228-3 - Introduction to Ring Theory. Math 228 Unless...

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Introduction to Ring Theory. Math 228 Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra, Second Edition. Homework 3 - due February 2, 2010 D.3(a): Prove that the following relation on the set R of real numbers is an equivalence relation: a b if and only if cos a = cos b . Solution: It is reflexive: cos a = cos a , therefore a a . It is symmetric: a b iff cos a = cos b , iff cos b = cos a iff b a . It is transitive: a b and b c , iff cos a = cos b and cos b = cos c . But that implies cos a = cos c , which implies a c . A relationship that is reflexive, symmetric, and transitive is an equivalence relation. 2.1.12: Which of the following congruences have solutions: (a) x 2 1 (mod 3) (b) x 2 2 (mod 7) (c) x 2 3 (mod 11) Solution: (a) Since x 0 (mod 3) x 2 0 (mod 3) x 1 (mod 3) x 2 1 (mod 3) x 2 (mod 3) x 2 1 (mod 3) then x 2 1 (mod 3) has a solution for x 1 or 2 (mod 3) (b) Since x 0 (mod 7) x 2 0 (mod 7) x 1 (mod 7)
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hwk228-3 - Introduction to Ring Theory. Math 228 Unless...

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