Solution

\(\bar 3 \xrightarrow{f} \bar 4\) with \(\{1 \mapsto 2, 2 \mapsto 2, 3\mapsto 4\}\),\(A_1 = \{1,2\}, A_2=\{2,3\}, B_1=\{1\}, B_2=\{1,4\}\)

1. \(f^*(B_1)=\varnothing, f^*(B_2)=\{4\}\)

2. \(f_!(A_1)=\{2\},f_!(A_2)=\{2,4\}\)

3. \(f_*(A_1)=\{1,2,3\},f_*(A_2)=\{1,3,4\}\)