hw2 - Math 113 Homework # 2, selected solutions Fraleigh...

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Math 113 Homework # 2, selected solutions Fraleigh 3.26: Suppose that φ : ( S, * ) ( S 0 , * 0 ) is an isomorphism of binary structures; prove that φ - 1 is an isomorphism of binary structures. Since φ φ - 1 = id S 0 and φ - 1 φ = id S and the identity is bijective, it follows from a basic lemma from class that φ - 1 is a bijection. So we just have to check that φ - 1 ( x * 0 y ) = φ - 1 ( x ) * φ - 1 ( y ) for all x,y S 0 . Since φ is injective, it is enough to check that φ of the left side of this equation equals φ of the right side, i.e. φ ( φ - 1 ( x * 0 y )) = φ ( φ - 1 ( x ) * φ - 1 ( y )) . (1) By the definition of φ - 1 , the left side of equation (1) is φ ( φ - 1 ( x * 0 y )) = x * 0 y. Since φ is an isomorphism and by the definition of φ - 1 , the right side of equation (1) is φ ( φ - 1 ( x ) * φ - 1 ( y )) = φ ( φ - 1 ( x )) * 0 φ ( φ - 1 ( y )) = x * 0 y. Hence equation (1) holds. Fraleigh 4.19: (a) Suppose
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This note was uploaded on 01/19/2010 for the course MATH 113 taught by Professor Ogus during the Fall '08 term at Berkeley.

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hw2 - Math 113 Homework # 2, selected solutions Fraleigh...

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