Midterm_Practice_sol_Sum2011

Midterm_Practice_sol_Sum2011 - Midterm Practice Problems 1...

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Midterm Practice Problems 1. Define the operation * on the set of all integers by: = * xy xy 2 . A. Is the operation * associative? B. Is the operation * commutative? C. Does there exist an identity element? D. If there is an identity element, which integers have inverses? E. Is the system consisting of the set of integers with the operation * a group?
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2. Define the function by : fZ Z x even () x odd x1 fx 2x = Find A. , when S={-2, 0, 2, 3, 5} , (() ) 1 fS f fS B. when T = {-2, -1, 0, 1, 2} ,( () ) 11 fT f −− C. Does f have a left inverse? D. Does f have a right inverse?
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3. The following relation R is defined on the set Z of all integers. xRy if and only if for some integer k. += 22 xy2 k Prove or disprove that R is an equivalence relation.
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4. Prove that the statement is true for all positive integers n. () ( ... 222 2 nn 1 2n 1 123 n 6 ) + + ++++ =
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5. Let G be a multiplicative group. Prove that the inverse of an element is unique.
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Midterm_Practice_sol_Sum2011 - Midterm Practice Problems 1...

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