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

\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
$p$&$q$&$r$&$\neg p$&$\neg q$&$p \lor q$&$\neg p \land \neg q$&$\varphi$&$\psi$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & \T& \T& \F& \T& \T& \T\\
\hline
\textbf{F} &\textbf{F} &\textbf{T} & \T& \T& \F& \T& \T& \T\\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & \T& \F& \T& \F& \F& \F\\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & \T& \F& \T& \F& \T& \T\\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & \F& \T& \T& \F& \F& \F\\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & \F& \T& \T& \F& \T& \T\\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & \F& \F& \T& \F& \F& \F\\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & \F& \F& \T& \F& \T& \T\\
\hline
\end{tabular}

\item Which of the formulas is satisfiable? \\
\quad Both are satisfiable.
\item Which of the formulas is valid? \\
\quad Neither are valid.
\item  Is $\varphi$ equivalent to $\psi$? \\
\quad They are semantically equivalent.
\item  Does $\varphi$ semantically entail $\psi$? \\
\quad Yes.
\item  Does $\psi$ semantically entail $\varphi$? \\
\quad Yes.

\end{enumerate}
