ahw2 - Math 3124 Thursday, September 8 Second Homework...

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Unformatted text preview: Math 3124 Thursday, September 8 Second Homework Solutions 1. Exercise 4.2 on page 28. Let S = { a , b , c } and let A = { α , β , γ , δ } , where α , β , γ , and δ are the elements of M ( S ) defined as follows. α ( a ) = a α ( b ) = b α ( c ) = c β ( a ) = b β ( b ) = a β ( c ) = c γ ( a ) = a γ ( b ) = a γ ( c ) = a δ ( a ) = b δ ( b ) = b δ ( c ) = b (a) ◦ α β γ δ α α β γ δ β β α δ γ γ γ γ γ γ δ δ δ δ δ (b) Composition of mappings is an associative operation (Theorem 4.1(a) in the book). (c) ◦ is not commutative; for example β ◦ γ = δ , yet γ ◦ β = γ . (d) The identity element for ◦ is α . 2. Problem 5.14 on page 34 Let H = { f : R → R | f ( x ) 6 = 0 for all x ∈ R } . For f , g ∈ H , define f g by ( f g )( x ) = f ( x ) g ( x ) for all x ∈ R . Then f g ∈ H . Verify that with this operation H is a group. How does this group differ from the group of invertible mappings in M ( R ) ? Also, is this group H abelian?...
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

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ahw2 - Math 3124 Thursday, September 8 Second Homework...

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