\item \self Consider the propositional formula $\varphi = (p \rightarrow q)
    \wedge(q \rightarrow r) \wedge (\neg r \vee p)$.

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

\begin{tabular}{|c|c|c||c|c|c|c|c|}
\hline
$p$&$q$&$r$&$(p \rightarrow q)$&$(q \rightarrow r)$&$\;\neg r\;$&$(\neg r \vee p)$&$\quad\varphi\quad$\\
\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 $\varphi$ satisfiable?
\item  Is $\varphi$ valid?
\item Give a formula $\psi$ that semantically entails $\varphi$
    (i.e., it should be the case that $\psi \models \varphi$).
\item How can you check, using a truth table, whether $\psi$
    semantically entails $\varphi$?

\end{enumerate}
