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Unformatted text preview: 32 14. October 18th: Isomorphisms 14.1 . Let F be a family of sets. Prove that the isomorphism relation = defines an equivalence relation on F . That is, show that (for all sets A , B , C in F ) A = A ; If A = B , then B = A ; If A = B and B = C , then A = C . Answer: A = A because 1 A : A A is an isomorphism. A = B = B = A : if f : A B is an isomorphism, then it has an inverse function f 1 : B A , which is also an isomorphism. If A = B and B = C , then there are isomorphisms f : A B and g : B C . The composition g f : A C is then an isomorphism (Exercise 12.3), proving that A = C . 14.2 . A certain function f : Z Z is defined by f ( x ) = ax + b , where a and b are integers. Assume that f is an isomorphism. What can you say about a and b ? What is the inverse of f ? Answer: If f is an isomorphism, it has an inverse g : Z Z . Since f g = 1 Z , we must have f g ( n ) = n for all n Z , that is: n = f g ( n ) = f ( g ( n )) = ag ( n ) + b . This says that g ( n ) = n b a . Thus a must be an integer such that ( n b ) /a is an integer for all n . It follows that a = 1. Thus, either f ( n ) = n + b and f 1 ( n ) = g ( n ) = n b or f ( n ) = n + b and f 1 ( n ) = g ( n ) = n + b . There is no restriction of what b may be. 14.3 . List all the elements of the set of bijections of A = { a, b,c, d } to itself, by indicating the image of each element. For example, a a b c c b d d denotes the bijection which swaps b and c . Next to each element, write its inverse. Find one element s that is not the identity and such that s s is the identity. Find one element t that is not the identity and such that t t t is the identity. Find one element u that is not the identity, such that u u is not the identity, and such that u u u u is the identity. 14. OCTOBER 18TH: ISOMORPHISMS 33 Answer: There are 24 distinct bijections: a a b b c c d d , a a b b c d d c , a a b c c b d d , a a b c c d d b , a a b d c b d c , a a b d c c d b a b b a c c d d , a b b a c d d c , a b b c c a d d , a b b c c d d a , a b b d c a d c , a b b d c c d a a c b a c b d d ,...
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 Fall '11
 Aluffi

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