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42 lines
1.3 KiB
42 lines
1.3 KiB
\item \self Consider the propositional formulas $\varphi = (p \lor q)
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\rightarrow r$, and $\psi = r \lor (\neg p \land \neg q)$.
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\begin{enumerate}
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\item Fill out the truth table for $\varphi$ and $\psi$ (and
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their subformulas).
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\begin{tabular}{|c|c|c||c|c|c|c||c|c|}
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\hline
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$p$&$q$&$r$&$\neg p$&$\neg q$&$p \lor q$&$\neg p \land \neg q$&$\varphi$&$\psi$\\
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\hline
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\hline
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\textbf{F} &\textbf{F} &\textbf{F} & \T& \T& \F& \T& \T& \T\\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} & \T& \T& \F& \T& \T& \T\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} & \T& \F& \T& \F& \F& \F\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} & \T& \F& \T& \F& \T& \T\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} & \F& \T& \T& \F& \F& \F\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} & \F& \T& \T& \F& \T& \T\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} & \F& \F& \T& \F& \F& \F\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} & \F& \F& \T& \F& \T& \T\\
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\hline
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\end{tabular}
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\item Which of the formulas is satisfiable? \\
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\quad Both are satisfiable.
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\item Which of the formulas is valid? \\
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\quad Neither are valid.
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\item Is $\varphi$ equivalent to $\psi$? \\
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\quad They are semantically equivalent.
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\item Does $\varphi$ semantically entail $\psi$? \\
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\quad Yes.
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\item Does $\psi$ semantically entail $\varphi$? \\
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\quad Yes.
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\end{enumerate}
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