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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/19/2010 for the course MATH 113 taught by Professor Ogus during the Fall '08 term at Berkeley.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online