\item \self Consider the propositional formula $\varphi = (\neg p \imp r) \land (r \imp \neg p) \land q$.

\begin{enumerate}
  \item Fill out the truth table for $\varphi$ (and its
      subformulas).

\begin{tabular}{|c|c|c||c|c|c|c|}
\hline
$p$&$q$&$r$&$\;\neg p\;$&$(\neg p \imp r)$&$(r \imp \neg p)$&$\quad\varphi\quad$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & \T & \F & \T & \F \\
\hline
\textbf{F} &\textbf{F} &\textbf{T} & \T & \T & \T & \F \\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & \T & \F & \T & \F \\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & \T & \T & \T & \T \\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & \F & \T & \T & \F \\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & \F & \T & \F & \F \\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & \F & \T & \T & \T \\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & \F & \T & \F & \F \\
\hline
\end{tabular}

\item Is $\varphi$ satisfiable? \\
\quad Yes.
\item Is $\varphi$ valid? \\
\quad No.
\item Give a formula $\psi$ that semantically entails $\varphi$. \\
\quad For any formula $\varphi$ it holds that $\bot \models \varphi$, we can therefore choose $\psi = \bot$. We could also represent $\varphi$ as DNF: $(\neg p \land q \land r) \lor (p \land q \land \neg r)$. This is an equivalent formula and therefore semantically entails $\varphi$.
\item Give a formula $\psi$ such that $\varphi$ semantically entails $\psi$. \\
\quad For any formula $\varphi$ it holds that $\varphi \models \top$. We could also again choose $\varphi$ in DNF.

\end{enumerate}