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38 lines
1.1 KiB
38 lines
1.1 KiB
\item \self Consider the propositional formula $\varphi = (p \rightarrow q)
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\wedge(q \rightarrow r) \wedge (\neg r \vee p)$.
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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|c|}
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\hline
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$p$&$q$&$r$&$(p \rightarrow q)$&$(q \rightarrow r)$&$\;\neg r\;$&$(\neg r \vee p)$&$\quad\varphi\quad$\\
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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 $\varphi$ satisfiable?
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\item Is $\varphi$ 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 How can you check, using a truth table, whether $\psi$
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semantically entails $\varphi$?
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\end{enumerate}
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