4023f10ps5a

4023f10ps5a - Homework #5 Due: October 13, 2010 1. Decide...

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Homework #5 Due: October 13, 2010 1. Decide whether each of the following sets forms a group with respect to the given operation. If it does form a group, give the identity element and the inverse of each element. If it does not form a group, identify which of the group axioms fail. (a) Q / = n m n : m; n 2 Z ; m 6 = 0 ; n 6 = 0 o with the operation of multiplication. Answer: Group. Identity = 1; ( m=n ) ¡ 1 = n=m . (b) 1 ; 1 g ‰ Z with the operation of multiplication. Answer: Group. Identity = 1; Each element is its own inverse. (c) Z with the operation of subtraction. Answer: Not a group. Subtraction is not associative: a ¡ ( b ¡ c ) 6 = ( a ¡ b ) ¡ c . (d) G = ‰• 1 n 0 1 : n 2 Z ± with the operation of matrix multiplication. Answer: Group. This set is closed under matrix multiplication 1 n 0 1 ‚• 1 m 0 1 = 1 n + m 0 1 and inverses 1 n 0 1 ¡ 1 = 1 ¡ n 0 1 . The identity is the identity ma- trix. (e) The set F of all functions from f 1 ; 2 ; 3 g to itself with the operation of composition of functions. Answer: Not a group. Not every function has an inverse. For example, the constant function f ( a ) = 1 for all a does not have an inverse function under composition.
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4023f10ps5a - Homework #5 Due: October 13, 2010 1. Decide...

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