Tarea 5

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\begin{document}

%TC:ignore
\begin{titlepage}
{\centering\LARGE\bf \thetitle}

Larry Murcia Terranova

%\bigskip

\end{titlepage}
% Resize BoxTable
\begin{center}
\begin{tabular}[c]{|c|c|}
\hline
\multirow{2}{8cm}{Cuantificaciones Sin Efecto} & $\exists x \ \varphi \ \equiv \ \varphi$ \ \ si $x\notin lib(\varphi) $ \\ 
& $\forall x \  \varphi \ \equiv \ \varphi$ \ \ si $x\notin lib(\varphi)$ \\
\hline
\multirow{2}{8cm}{Cuantificaciones Sucesivas} & $\exists x \exists y \ \varphi \ \equiv \ \exists y \exists x \ \varphi$ \\
& $\forall x \forall y \ \varphi \ \equiv \ \forall y \forall x \ \varphi$ \\
\hline
\multirow{2}{8cm}{Negaci\'on De Cuantificaciones} & $\lnot\forall x \ \varphi \ \equiv \ \exists x \ \lnot\varphi$ \\
& $\lnot\exists x \ \varphi \ \equiv \ \forall x \ \lnot\varphi$ \\
\hline
\multirow{6}{8cm}{Cuantificaci\'on Conjunci\'on y Disyunci\'on} & $\exists x \ \varphi \vee \exists x \ \Psi \ \equiv \ \exists x \ (\varphi \vee \Psi)$ \\
& $\forall x \ \varphi \wedge \forall x \ \Psi \ \equiv \ \forall x \ (\varphi \wedge \Psi)$ \\
& $\exists x \ \varphi \vee \Psi \ \equiv \ \exists x \ (\varphi \vee \Psi)$ \ \ si $x \notin lib(\Psi)$ \\
& $\exists x \ \varphi \wedge \Psi \ \equiv \ \exists x \ (\varphi \wedge \Psi)$ \ \ si $x \notin lib(\Psi)$ \\
& $\forall x \ \varphi \vee \Psi \ \equiv \ \forall x \ (\varphi \vee \Psi)$ \ \ si $x \notin lib(\Psi)$ \\
& $\forall x \ \varphi \wedge \Psi \ \equiv \ \forall x \ (\varphi \wedge \Psi)$ \ \ si $x \notin lib(\Psi)$ \\
\hline
\multirow{4}{8cm}{Cuantificadores En Condicionales} & $\exists x \ \varphi\Rightarrow\Psi \ \equiv \ \forall x \ (\varphi\Rightarrow\Psi)$ \ \ si $x \notin lib(\Psi)$ \\
& $\forall x \ \varphi\Rightarrow\Psi \ \equiv \ \exists x \ (\varphi\Rightarrow\Psi)$ \ \ si $x \notin lib(\Psi)$ \\
& $\varphi\Rightarrow\exists x \ \Psi \ \equiv \ \exists x (\varphi\Rightarrow\Psi)$ \ \ si $x \notin lib(\varphi)$ \\
& $\varphi\Rightarrow\forall x \ \Psi \ \equiv \ \forall x (\varphi\Rightarrow\Psi)$ \ \ si $x \notin lib(\varphi)$ \\
\hline
\multirow{2}{8cm}{Renombramiento De Variable} & $\exists x \ \varphi \ \equiv \ \exists y \ \varphi_{[x/y]}$ \ \ si $y\neq x$ \ y \ $y \notin lib(\varphi)$ \\
& $\forall x \ \varphi \ \equiv \ \forall y \ \varphi_{[x/y]}$ \ \ si $y\neq x$ \ y \ $y \notin lib(\varphi)$ \\
\hline
\end{tabular}
\end{center}

\end{document}
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