SampleExam1 - MATH 373: Sample Exam 1 Monday, February 15,...

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Unformatted text preview: MATH 373: Sample Exam 1 Monday, February 15, 2010 Answer all questions in the answer booklet. Please do these in order. You are expected to justify your answers in a manner that an average MATH 373 student should be able to follow. This includes sentences for clarification. Point values are on the right hand side. They (should) sum up to 100. 1. For x,y Z , define x # y = x + y + 2 and x * y = 2 x + 2 y . (Here + denotes ordinary addition of integers and 2 x , 2 y are the ordinary products of 2 with x and y .) (a) Is Z a group with #? Justify your answer by showing that # satisfies all the group axioms or by showing that it fails to satisfy at least one axiom. [12 pts] (b) Is Z a group with * ? Justify your answer by showing that * satisfies all the group axioms or by showing that it fails to satisfy at least one axiom. [12 pts] 2. Let R 2 be the group of all ordered pairs of real numbers with vector addition. That is, R 2 = { ( x,y ) : x,y R } and ( x,y ) + ( u,v ) = (...
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