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\item \ifassignmentsheet \points{2} \else \prac \fi Consider the propositional formula $\phi = (p \lor \lnot q) \imp (\lnot p \land \lnot r)$. Fill out the truth table for $\phi$
and its subformulas. Compute a CNF as well as a DNF for $\phi$ from
the truth table.
\begin{tabular}{|c|c|c||c|c|c|c|c||c|}
\hline
$p$ & $q$ & $r$ & $\lnot q$ & $p \lor \lnot q$ & $\lnot p$ & $\lnot r$ & $\lnot p \land \lnot r$ & $\phi = (p \lor \lnot q) \imp (\lnot p \land \lnot r)$\\
\hline
\hline%p q r %!q !p !r phi
\F &\F &\F & \T&\T &\T &\T &\T &\T\\ \hline
\F &\F &\T & \T&\T &\T &\F &\F &\F\\ \hline
\F &\T &\F & \F&\F &\T &\T &\T &\T\\ \hline
\F &\T &\T & \F&\F &\T &\F &\F &\T\\ \hline
\T &\F &\F & \T&\T &\F &\T &\F &\F\\ \hline
\T &\F &\T & \T&\T &\F &\F &\F &\F\\ \hline
\T &\T &\F & \F&\T &\F &\T &\F &\F\\ \hline
\T &\T &\T & \F&\T &\F &\F &\F &\F\\ \hline
\end{tabular}
The resulting CNF:
\begin{align*}
(\clause{p;q;\lnot r})&~\land \\
(\clause{\lnot p;q;r})&~\land \\
(\clause{\lnot p;q;\lnot r})&~\land \\
(\clause{\lnot p;\lnot q;r})&~\land \\
(\clause{\lnot p;\lnot q;\lnot r})&
\end{align*}
The resulting DNF:
\begin{align*}
(\cube{\lnot p;\lnot q;\lnot r})&~\lor \\
(\cube{\lnot p; q;\lnot r})&~\lor \\
(\cube{\lnot p; q; r})&
\end{align*}