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39 lines
1.2 KiB
39 lines
1.2 KiB
\item \self Consider the propositional formula $\varphi = p \imp
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(q \imp r)$.
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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|}
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\hline
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$p$&$q$&$r$&$(q \imp r)$&$\varphi=p \imp (q \imp r)$\\
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\hline
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\hline
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\textbf{F} &\textbf{F} &\textbf{F} & \T & \T \\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} & \T & \T \\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} & \F & \T \\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} & \T & \T \\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} & \T & \T \\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} & \T & \T \\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} & \F & \F \\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} & \T & \T \\
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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 Give a formula $\psi$ that is semantically equivalent to
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$\varphi$, but does not use the ``$\imp$'' connective.
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\quad $\psi = \neg p \lor (\neg q \lor r)$
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\item How can you check whether $\psi$ is semantically equivalent
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to $\varphi$?
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\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.
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\end{enumerate}
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