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35 lines
1.1 KiB
35 lines
1.1 KiB
\item \self Consider the propositional formula
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$\varphi = ((p \rightarrow q)\wedge(\neg p \rightarrow \neg q)) \rightarrow r$.
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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|c|c|c|}
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\hline
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$p$&$q$&$r$&$\;\neg p\;$&$\;\neg q\;$&$(p \rightarrow q)$&$(\neg p \rightarrow \neg q)$&$(p \rightarrow q)\wedge (\neg p \rightarrow \neg q)$&$\quad\varphi\quad$\\
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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 $\varphi$ unsatisfiable?
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\item Is the negation of $\varphi$ valid?
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\item Give a formula $\psi$ that is semantically equivalent to
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$\varphi$, but does not use the ``$\rightarrow$'' connective.
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\end{enumerate}
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