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Quiz 2
1. If $T$ is a tautology, show that $P \wedge T \equiv P$.
Solution: It suffices to show that $(P \wedge T) \leftrightarrow P$ is a tautology. Compute
$$\begin{array}{|l|l|l|l|l|}
\hline
P & T & P \wedge T & (P \wedge T) \leftrightarrow P \\
\hline
1&1&1&1 \\
0&1&0&1 \\
\hline
\end{array}$$
2. Show that $\neg (F \vee G) \equiv (\neg F) \wedge (\neg G)$.
Solution: It suffices to show that $\neg (F \vee G) \leftrightarrow ((\neg F) \wedge (\neg G))$ is a tautology. Compute
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline
F & G & \neg F & \neg G & F \vee G & \neg (F \vee G) & (\neg F) \wedge (\neg G) & (\neg (F \vee G)) \leftrightarrow ((\neg F) \wedge (\neg G))\\
\hline
1&1&0&0&1&0&0&1 \\
1&0&0&1&1&0&0&1 \\
0&1&1&0&1&0&0&1 \\
0&0&1&1&0&1&1&1 \\
\hline
\end{array}$$
