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\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}
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