You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
36 lines
1.3 KiB
36 lines
1.3 KiB
\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*}
|