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37 lines
1.2 KiB
37 lines
1.2 KiB
\item \self Consider the Boolean functions $\varphi_1$ and $\varphi_2$ over variables $p$,
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$q$, and $r$. Their truth table is given below.
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\begin{tabular}{|c|c|c|c|c||c|c|}
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\hline
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$p$&$q$&$r$&$\varphi_1$&$\varphi_2$&$\psi$&$\gamma$\\
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\hline
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\hline
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\textbf{F} &\textbf{F} &\textbf{F} &\textbf{F} &\textbf{F} & &\\
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\hline
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\textbf{F} &\textbf{F} &\textbf{T} &\textbf{F} &\textbf{F} & &\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{F} &\textbf{T} &\textbf{T} & &\\
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\hline
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\textbf{F} &\textbf{T} &\textbf{T} &\textbf{F} &\textbf{F} & &\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{F} &\textbf{T} &\textbf{F} & &\\
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\hline
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\textbf{T} &\textbf{F} &\textbf{T} &\textbf{F} &\textbf{T} & &\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{F} &\textbf{F} &\textbf{F} & &\\
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\hline
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\textbf{T} &\textbf{T} &\textbf{T} &\textbf{F} &\textbf{F} & &\\
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\hline
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\end{tabular}
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\begin{enumerate}
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\item Fill the column for $\psi$ such that $\varphi_1$ entails
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$\psi$ (i.e., $\varphi_1 \models \psi$), but $\varphi_2$ does
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\emph{not} entail $\psi$ (i.e., $\varphi_2 \nvDash \psi$).
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\item Fill the column for $\gamma$ such that $\varphi_1$ implies
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$\gamma$ (i.e., $\varphi_1 \rightarrow \gamma$) as well as $\varphi_2$ implies $\gamma$ (i.e., $\varphi_2 \rightarrow \gamma$).
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\end{enumerate}
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