Choose sets \(X,Y\) with between two and four elements each, and choose a function \(X \xrightarrow{f} Y\) NOCARD
Choose two different subsets \(B_1, B_2 \subseteq Y\) and find \(f^*(B_1)\) and \(f^*(B_2)\)
Choose two different subsets \(A_1, A_2 \subseteq X\) and find \(f_!(A_1)\) and \(f_!(A_2)\)
With the same \(A_1, A_2\) find \(f_*(A_1)\) and \(f_*(A_2)\)
\(\bar 3 \xrightarrow{f} \bar 4\) with \(\{1 \mapsto 2, 2 \mapsto 2, 3\mapsto 4\}\),\(A_1 = \{1,2\}, A_2=\{2,3\}, B_1=\{1\}, B_2=\{1,4\}\)
\(f^*(B_1)=\varnothing, f^*(B_2)=\{4\}\)
\(f_!(A_1)=\{2\},f_!(A_2)=\{2,4\}\)
\(f_*(A_1)=\{1,2,3\},f_*(A_2)=\{1,3,4\}\)