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

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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 ) = 0 for all x R } . For f , g H , define fg by ( fg )( x ) = f ( x ) g ( x ) for all x R . Then fg 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? First note that we do indeed have a binary operation, because if f , g H , then fg : R R , and fg ( x ) = f ( x ) g ( x ) = 0 for all x R . We now have the three axioms for a group to check.
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