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41 lines
1.4 KiB
41 lines
1.4 KiB
\item \self Consider the propositional formula $\psi = (p \imp q)
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\land(q \imp r) \land (\neg r \lor p)$.
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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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$p$&$q$&$r$&$(p \imp q)$&$(q \imp r)$&$\;\neg r\;$&$(\neg r \lor p)$&$\quad\psi\quad$\\
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\hline
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\hline
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\textbf{F} &\textbf{F} &\textbf{F} & \T & \T & \T & \T & \T\\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} & \T & \T & \F & \F & \F\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} & \T & \T & \F & \F & \F\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} & \F & \T & \T & \T & \F\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} & \F & \T & \F & \T & \F\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} & \T & \T & \F & \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 Is $\varphi$ valid? \\
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\quad No.
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\item Give a formula $\varphi$ that semantically entails $\psi$. \\
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\quad An equivalent formula $\varphi$ semantically entails $\psi$, therefore we let $\varphi = \psi$.
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\item How can you check, using a truth table, whether $\varphi$
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semantically entails $\psi$? \\
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\quad We can check this by looking at the models that satisfy the two formulas. If every model that satisfies $\varphi$ also models $\psi$, then $\varphi$ semantically entails $\psi$.
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\end{enumerate}
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