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\item \self Consider the propositional formula $\varphi = (\neg p \rightarrow r) \wedge (r \rightarrow \neg p) \wedge q$.
\begin{enumerate}
\item Fill out the truth table for $\varphi$ (and its
subformulas).
\begin{tabular}{|c|c|c||c|c|c|c|}
\hline
$p$&$q$&$r$&$\;\neg p\;$&$(\neg p \rightarrow r)$&$(r \rightarrow \neg p)$&$\quad\varphi\quad$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & & & &\\
\hline
\textbf{F} &\textbf{F} &\textbf{T} & & & &\\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & & & &\\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & & & &\\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & & & &\\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & & & &\\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & & & &\\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & & & &\\
\hline
\end{tabular}
\item Is the negation of $\varphi$ satisfiable?
\item Is the negation of $\varphi$ valid?
\item Give a formula $\psi$ that semantically entails $\varphi$
(i.e., it should be the case that $\psi \models \varphi$).
\item Give a formula $\psi$ such that $\varphi$ semantically entails $\psi$
(i.e., it should be the case that $\varphi \models \psi$).
\end{enumerate}