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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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Johnson
 Math, Algebra

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