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Exercise 29 - Chapter 3

edited June 2018 in Exercises
  1. What is the inverse \( f^{−1} : 3 \rightarrow A \) of the function \(f\) given below?

  2. How many distinct isomorphisms are there \( A \rightarrow 3 \)?

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The set \( A := \{a, b, c \} \) and the set \( \underline{3} = \{1, 2, 3 \} \) are isomorphic; that is, there exists an isomorphism \( f : A \rightarrow \underline{3} \) given by \( f(a) = 2, f(b) = 1, f(c) = 3 \).

Comments

  • 1.
    1. \(f^{-1}(2)=a,f^{-1}(1)=b,f^{-1}(3)=c\)

    2. There is one for each permutation on three elements, so there are 3!=6.

    Comment Source:1. \\(f^{-1}(2)=a,f^{-1}(1)=b,f^{-1}(3)=c\\) 2. There is one for each permutation on three elements, so there are 3!=6.
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