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\item \self Consider the domain $A=\{Asparagus, \;Bell \; Pepper, \;Cabbage, \;Tomato, \\ \; Onion, \;Zucchini, \;Eggplant, \; Mushroom\}$ and the two different symbolic encodings for $A$ given below. Which one gives a shorter characteristic function for the following sets? \begin{enumerate} \item $B = \{Asparagus, \; Mushroom\}$ \item $C = \{Bell \; Pepper, \; Onion, \; Zucchini, \; Eggplant\}$ \item $D = \{Cabbage, \; Tomato\}$ \end{enumerate} Illustrate your answer by giving the characteristic function for $B$, $C$ and $D$ in both encodings.
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\begin{tabular}{l|l|l|l} \hline \multicolumn{3}{c}{\textbf{Encoding 1}} \\ \hline Element & x & y & z\\ \hline Asparagus & $0$ & $0$ & $0$ \\ Bell Pepper & $0$ & $0$ & $1$ \\ Cabbage & $0$ & $1$ & $0$ \\ Tomato & $0$ & $1$ & $1$ \\ Onion & $1$ & $0$ & $0$ \\ Zucchini & $1$ & $0$ & $1$ \\ Eggplant & $1$ & $1$ & $0$ \\ Mushroom & $1$ & $1$ & $1$ \end{tabular} %\end{minipage}
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\hspace{3cm} \begin{tabular}{l|l|l|l} \hline \multicolumn{3}{c}{\textbf{Encoding 2}} \\ \hline Element & x & y & z \\ \hline Asparagus & $1$ & $1$ & $0$ \\ Bell Pepper & $1$ & $0$ & $1$ \\ Cabbage & $0$ & $1$ & $0$ \\ Tomato & $1$ & $1$ & $1$ \\ Onion & $0$ & $0$ & $0$ \\ Zucchini & $0$ & $0$ & $1$ \\ Eggplant & $1$ & $0$ & $0$ \\ Mushroom & $0$ & $1$ & $1$ \end{tabular} %\end{minipage}
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