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Unformatted text preview: Math 113 Section 5 Midterm #1 Solutions Fall 2010 Wednesday, September 22 1. Determine which of the following sets with 2to1 operations form groups. For those that are groups, prove that they are groups. For those that are not groups, describe why they are not groups. (a) ( Z , ) is not a group because not every integer has a multiplicative inverse which is also an integer. For example, 1 2 / Z . (b) ( Q > , ) is a group. Closure: If a b > 0 and a b > 0 then aa bb > 0. Associativity follows from associativity of multiplication in R . Identity: 1 Q > . Inverses: If a b > 0 then b a > 0 and a b b a = 1. (c) ( { M GL ( n, R )  M is diagonal } , ) is a group. Closure: Let A and B be diagonal n n matrices with entries a 1 ,a 2 ,...,a n and b 1 ,b 2 ,...,b n respectively. Then AB is diagonal with entries a 1 b 1 ,a 2 b 2 ,...a n b n . Associativity follows from associativity of matrix multiplication. Note that for n n matrices, ( AB ) C and A ( BC ) are always defined.) are always defined....
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This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.
 Summer '11
 gelfand
 Chemistry

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