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

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

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

\item Is $\varphi$ satisfiable? \\
\quad Yes.
\item Give a formula $\psi$ that is semantically equivalent to
    $\varphi$, but does not use the ``$\imp$'' connective.
\quad $\psi = \neg p \lor (\neg q \lor r)$
\item How can you check whether $\psi$ is semantically equivalent
    to $\varphi$?
\quad Since both formulas are relatives compact, we can use their respective truth table to check whether they are semantically equivalent. We do this by checking whether they evaluate to $\T$ under the same models.
\end{enumerate}
