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39 lines
1.5 KiB
39 lines
1.5 KiB
\item \self Consider the propositional formula $\varphi = (\neg p \imp r) \land (r \imp \neg p) \land q$.
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\begin{enumerate}
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\item Fill out the truth table for $\varphi$ (and its
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subformulas).
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\begin{tabular}{|c|c|c||c|c|c|c|}
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\hline
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$p$&$q$&$r$&$\;\neg p\;$&$(\neg p \imp r)$&$(r \imp \neg p)$&$\quad\varphi\quad$\\
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\hline
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\hline
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\textbf{F} &\textbf{F} &\textbf{F} & \T & \F & \T & \F \\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} & \T & \T & \T & \F \\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} & \T & \F & \T & \F \\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} & \T & \T & \T & \T \\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} & \F & \T & \T & \F \\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} & \F & \T & \F & \F \\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} & \F & \T & \T & \T \\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} & \F & \T & \F & \F \\
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\hline
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\end{tabular}
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\item Is $\varphi$ satisfiable? \\
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\quad Yes.
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\item Is $\varphi$ valid? \\
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\quad No.
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\item Give a formula $\psi$ that semantically entails $\varphi$. \\
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\quad For any formula $\varphi$ it holds that $\bot \models \varphi$, we can therefore choose $\psi = \bot$. We could also represent $\varphi$ as DNF: $(\neg p \land q \land r) \lor (p \land q \land \neg r)$. This is an equivalent formula and therefore semantically entails $\varphi$.
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\item Give a formula $\psi$ such that $\varphi$ semantically entails $\psi$. \\
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\quad For any formula $\varphi$ it holds that $\varphi \models \top$. We could also again choose $\varphi$ in DNF.
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\end{enumerate}
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