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\item \self Consider the propositional formula $\varphi = p \imp
(q \imp r)$.
\begin{enumerate} \item Fill out the truth table for $\varphi$ and its subformulas.
\begin{tabular}{|c|c|c||c|c|c|}\hline$p$&$q$&$r$&$(q \imp r)$&$\varphi=p \imp (q \imp r)$\\\hline\hline\textbf{F} &\textbf{F} &\textbf{F} & \T & \T \\\hline\textbf{F} &\textbf{F} &\textbf{T} & \T & \T \\\hline\textbf{F} &\textbf{T} &\textbf{F} & \F & \T \\\hline\textbf{F} &\textbf{T} &\textbf{T} & \T & \T \\\hline\textbf{T} &\textbf{F} &\textbf{F} & \T & \T \\\hline\textbf{T} &\textbf{F} &\textbf{T} & \T & \T \\\hline\textbf{T} &\textbf{T} &\textbf{F} & \F & \F \\\hline\textbf{T} &\textbf{T} &\textbf{T} & \T & \T \\\hline\end{tabular}
\item Is $\varphi$ satisfiable? \\\quad Yes.\item Give a formula $\psi$ that is semantically equivalent to $\varphi$, but does not use the ``$\imp$'' connective.\quad $\psi = \neg p \lor (\neg q \lor r)$\item How can you check whether $\psi$ is semantically equivalent to $\varphi$?\quad Since both formulas are relatives compact, we can use their respective truth table to check whether they are semantically equivalent. We do this by checking whether they evaluate to $\T$ under the same models.\end{enumerate}
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