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37 lines
1.1 KiB
37 lines
1.1 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} & & & &\\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} & & & &\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} & & & &\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} & & & &\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} & & & &\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} & & & &\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} & & & &\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} & & & &\\
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\hline
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\end{tabular}
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\item Is the negation of $\varphi$ satisfiable?
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\item Is the negation of $\varphi$ valid?
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\item Give a formula $\psi$ that semantically entails $\varphi$
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(i.e., it should be the case that $\psi \models \varphi$).
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\item Give a formula $\psi$ such that $\varphi$ semantically entails $\psi$
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(i.e., it should be the case that $\varphi \models \psi$).
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\end{enumerate}
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