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\item \self Consider the propositional formula $\varphi = (p \rightarrow q)
\wedge(q \rightarrow r) \wedge (\neg r \vee p)$.
\begin{enumerate} \item Fill out the truth table for $\varphi$ (and its subformulas).
\begin{tabular}{|c|c|c||c|c|c|c|c|}\hline$p$&$q$&$r$&$(p \rightarrow q)$&$(q \rightarrow r)$&$\;\neg r\;$&$(\neg r \vee p)$&$\quad\varphi\quad$\\\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 Is $\varphi$ satisfiable?\item Is $\varphi$ valid?\item Give a formula $\psi$ that semantically entails $\varphi$ (i.e., it should be the case that $\psi \models \varphi$).\item How can you check, using a truth table, whether $\psi$ semantically entails $\varphi$?
\end{enumerate}
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