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\begin{frame}
\frametitle{Logic Equivalence}

\begin{goal}{}
Formulas $\phi$ and $\psi$ are \emph{logically equivalent}, denoted
\begin{talign}
\phi \;\equiv\; \psi\;,
\end{talign}
if $\phi$ and $\psi$ have the same truth table.
\end{goal}
\smallskip

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\thd $p$ & \thd $q$ & \thd $\neg p$ & \thd $\neg p \vee q$ & \thd $p \to q$ \\
\hline
$\F$ & $\F$ & $\T$ & \malert{1}{2}{$\T$} & \malert{1}{2}{$\T$}\\
\hline
$\F$ & $\T$ & $\T$ & \malert{1}{2}{$\T$} & \malert{1}{2}{$\T$}\\
\hline
$\T$ & $\F$ & $\F$ & \malert{1}{2}{$\F$} & \malert{1}{2}{$\F$}\\
\hline
$\T$ & $\T$ & $\F$ & \malert{1}{2}{$\T$} & \malert{1}{2}{$\T$}\\
\hline
\end{tabular}\\
\medskip
\mpause[2]{\quad\quad \alert{$\neg p \vee q \;\equiv\; p \to q$}}
\end{center}
\vspace{-1ex}
\pause[4]

\begin{exampleblock}{More examples}
\begin{malign}
p \wedge q \;&\equiv\; \neg (\neg p \vee \neg q)
\\
p \vee q \;&\equiv\; q \vee p
\\
p \to \neg q \;&\equiv\; q \to \neg p
\end{malign}
\end{exampleblock}
\end{frame}