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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
 Isomorphism, Clockwise rotation, Bijection, Isomorphisms

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