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\item \lect Consider the propositional formulas $\varphi = (p \rightarrow
q) \vee \neg r$ and $\psi = (\neg r \wedge p) \vee (\neg q \rightarrow \neg r)$.
\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|c|}\hline$p$&$q$&$r$&$\neg q$&$\neg r$&$p \rightarrow q$&$\neg r \wedge p$&$\neg q \rightarrow \neg r$&$\varphi$&$\psi$\\\hline\hline\textbf{F} &\textbf{F} &\textbf{F} & T & T & T & F & T & T & T \\\hline \textbf{F} &\textbf{F} &\textbf{T} & T & F & T & F & F & T & F \\\hline\textbf{F} &\textbf{T} &\textbf{F} & F & T & T & F & T & T & T \\\hline\textbf{F} &\textbf{T} &\textbf{T} & F & F & T & F & T & T & T \\\hline\textbf{T} &\textbf{F} &\textbf{F} & T & T & F & T & T & T & T \\\hline\textbf{T} &\textbf{F} &\textbf{T} & T & F & F & F & F & F & F \\\hline\textbf{T} &\textbf{T} &\textbf{F} & F & T & T & T & T & T & T \\\hline\textbf{T} &\textbf{T} &\textbf{T} & F & F & T & F & T & T & T \\\hline\end{tabular}\vspace{0.5cm}
\item Which of the formulas is satisfiable?
Both of them are satisfiable.
\item Which of the formulas is valid?
None of them are valid.
\item Which of the two formulas $\varphi$ and $\psi$ entails the other?
It holds that $\psi \models \varphi$.
\end{enumerate}
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