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\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}
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