\item \self Consider the propositional formulas $\varphi = (p \vee q)
    \rightarrow r$, and $\psi = r \vee (\neg p \wedge \neg q)$.

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

\begin{tabular}{|c|c|c||c|c|c|c||c|c|}
\hline
$p$&$q$&$r$&$\neg p$&$\neg q$&$p \vee q$&$\neg p \wedge \neg q$&$\varphi$&$\psi$\\
\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 Which of the formulas is satisfiable?
\item  Which of the formulas is valid?
\item  Is $\varphi$ equivalent to $\psi$?
\item  Does $\varphi$ semantically entail $\psi$?
\item  Does $\psi$ semantically entail $\varphi$?

\end{enumerate}
