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\item \self Consider the propositional formula $\psi = (p \imp q)
\land(q \imp r) \land (\neg r \lor p)$.
\begin{enumerate} \item Fill out the truth table for $\varphi$ (and its subformulas).
\begin{tabular}{|c|c|c||c|c|c|c|c|}\hline$p$&$q$&$r$&$(p \imp q)$&$(q \imp r)$&$\;\neg r\;$&$(\neg r \lor p)$&$\quad\psi\quad$\\\hline\hline\textbf{F} &\textbf{F} &\textbf{F} & \T & \T & \T & \T & \T\\\hline\textbf{F} &\textbf{F} &\textbf{T} & \T & \T & \F & \F & \F\\\hline\textbf{F} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\\hline\textbf{F} &\textbf{T} &\textbf{T} & \T & \T & \F & \F & \F\\\hline\textbf{T} &\textbf{F} &\textbf{F} & \F & \T & \T & \T & \F\\\hline\textbf{T} &\textbf{F} &\textbf{T} & \F & \T & \F & \T & \F\\\hline\textbf{T} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\\hline\textbf{T} &\textbf{T} &\textbf{T} & \T & \T & \F & \T & \T\\\hline\end{tabular}
\item Is $\varphi$ satisfiable? \\\quad Yes.\item Is $\varphi$ valid? \\\quad No.\item Give a formula $\varphi$ that semantically entails $\psi$. \\\quad An equivalent formula $\varphi$ semantically entails $\psi$, therefore we let $\varphi = \psi$.\item How can you check, using a truth table, whether $\varphi$ semantically entails $\psi$? \\\quad We can check this by looking at the models that satisfy the two formulas. If every model that satisfies $\varphi$ also models $\psi$, then $\varphi$ semantically entails $\psi$.
\end{enumerate}
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