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What is the inverse \( f^{−1} : 3 \rightarrow A \) of the function \(f\) given below?

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

## Comments

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

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

`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.`