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Exercise 29 - Chapter 3

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

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

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 $$f(a) = 2, f(b) = 1, f(c) = 3$$.

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