pomucka.tex

%
%  Pomůcka do LZUI
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%  Created by Milan Vít on 2010-11-02.
%  Copyright (c) 2010 Mion Studios. All rights reserved.
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\begin{document}

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\section{Logické funkce}

\begin{tabular}{c|c||c|c|c|c}
	$a$	& $b$	& $\,\&\,$	& $\vee$	& $\rightarrow$	& $\leftrightarrow$	\\ \hline\hline
	0	& 0		& 0			& 0			& 1				& 1					\\ \hline
	0	& 1		& 0			& 1			& 1				& 0					\\ \hline
	1	& 0		& 0			& 1			& 0				& 0					\\ \hline
	1	& 1		& 1			& 1			& 1				& 1					\\
\end{tabular}

\section{Dualita}

\begin{tabular}{c|c||c|c}
	$a$	& $b$	& $(a \,\&\, b)$	& $(a \,\&\, b)^D$	\\ \hline\hline
	0	& 0		& 0					& 1				\\ \hline
	0	& 1		& 0					& 1				\\ \hline
	1	& 0		& 0					& 1				\\ \hline
	1	& 1		& 1					& 0				\\
\end{tabular}

\section{Tablová pravidla}

\begin{tabular}{c|c|c||c|c|c}
	$\alpha$										& $\alpha_1$								& $\alpha_2$								& $\beta$												& $\beta_1$											& $\beta_2$											\\ \hline\hline
	$\neg\neg\mathcal{A}$							& $\mathcal{A}$								& 											& $\mathcal{B}_1 \vee \mathcal{B}_2$					& $\mathcal{B}_1$									& $\mathcal{B}_2$									\\ \hline
	$\mathcal{A}_1 \,\&\, \mathcal{A}_2$			& $\mathcal{A}_1$							& $\mathcal{A}_2$							& $\neg(\mathcal{B}_1 \,\&\, \mathcal{B}_2)$			& $\neg\mathcal{B}_1$								& $\neg\mathcal{B}_2$								\\ \hline
	$\neg(\mathcal{A}_1 \vee \mathcal{A}_2)$		& $\neg\mathcal{A}_1$						& $\neg\mathcal{A}_2$						& $\mathcal{B}_1 \rightarrow \mathcal{B}_2$				& $\neg\mathcal{B}_1$								& $\mathcal{B}_2$									\\ \hline
	$\neg(\mathcal{A}_1 \rightarrow \mathcal{A}_2)$	& $\mathcal{A}_1$							& $\neg\mathcal{A}_2$						& $\mathcal{B}_1 \leftarrow \mathcal{B}_2$				& $\mathcal{B}_1$									& $\neg\mathcal{B}_2$								\\ \hline
	$\neg(\mathcal{A}_1 \leftarrow \mathcal{A}_2)$	& $\neg\mathcal{A}_1$						& $\mathcal{A}_2$							& $\neg(\mathcal{B}_1 \leftrightarrow \mathcal{B}_2)$	& $\neg(\mathcal{B}_1 \rightarrow \mathcal{B}_2)$	& $\neg(\mathcal{B}_1 \leftarrow \mathcal{B}_2)$	\\ \hline
	$\mathcal{A}_1 \leftrightarrow \mathcal{A}_2$	& $\mathcal{A}_1 \rightarrow \mathcal{A}_2$	& $\mathcal{A}_1 \leftarrow \mathcal{A}_2$	& 														& 													& 													\\
\end{tabular}

\section{Ekvivalence výrokových formulí}

%\begin{eqnarray*}
$(\mathcal{X} \,\&\, \mathcal{X}) \leftrightarrow \mathcal{X}$																																			\\
$(\mathcal{X} \vee \mathcal{X}) \leftrightarrow \mathcal{X}$																																			\\
$(\mathcal{X} \,\&\, \mathcal{Y}) \leftrightarrow (\mathcal{Y} \,\&\, \mathcal{X})$																														\\
$(\mathcal{X} \vee \mathcal{Y}) \leftrightarrow (\mathcal{Y} \vee \mathcal{X})$																															\\
$\left(\left(\mathcal{X} \,\&\, \mathcal{Y}\right) \,\&\, \mathcal{Z}\right) \leftrightarrow \left(\mathcal{X} \,\&\, \left(\mathcal{Y} \,\&\, \mathcal{Z}\right)\right) $								\\
$\left(\left(\mathcal{X} \vee \mathcal{Y}\right) \vee \mathcal{Z}\right) \leftrightarrow \left(\mathcal{X} \vee \left(\mathcal{Y} \vee \mathcal{Z}\right)\right) $										\\
$\left(\left(\mathcal{X} \,\&\, \mathcal{Y}\right) \vee \mathcal{Z}\right) \leftrightarrow \left(\left(\mathcal{X} \vee \mathcal{Y}\right) \,\&\, \left(\mathcal{Y} \vee \mathcal{Z}\right)\right)$		\\
$\left(\left(\mathcal{X} \vee \mathcal{Y}\right) \,\&\, \mathcal{Z}\right) \leftrightarrow \left(\left(\mathcal{X} \,\&\, \mathcal{Y}\right) \vee \left(\mathcal{Y} \,\&\, \mathcal{Z}\right)\right)$	\\
$(\mathcal{X} \vee \neg{}\mathcal{X}) \leftrightarrow \text{true}$																																		\\
$(\mathcal{X} \,\&\, \neg{}\mathcal{X}) \leftrightarrow \text{false}$																																	\\
$\mathcal{X} \leftrightarrow \mathcal{X}$																																								\\
$\neg(\neg{}\mathcal{X}) \leftrightarrow \mathcal{X}$																																					\\
$\neg(\mathcal{X} \,\&\, \mathcal{Y}) \leftrightarrow (\neg{}\mathcal{X} \vee \neg{}\mathcal{Y})$																										\\
$\neg(\mathcal{X} \vee \mathcal{Y}) \leftrightarrow (\neg{}\mathcal{X} \,\&\, \neg{}\mathcal{Y})$																										\\
$\left(\mathcal{X} \vee \left(\mathcal{X} \,\&\, \mathcal{Y}\right)\right) \leftrightarrow \mathcal{X}$																									\\
$\left(\mathcal{X} \,\&\, \left(\mathcal{X} \vee \mathcal{Y}\right)\right) \leftrightarrow \mathcal{X}$																									\\
$(\mathcal{X} \vee \text{true}) \leftrightarrow \text{true}$																																			\\
$(\mathcal{X} \vee \text{false}) \leftrightarrow \mathcal{X}$																																			\\
$(\mathcal{X} \,\&\, \text{true}) \leftrightarrow \mathcal{X}$																																			\\
$(\mathcal{X} \,\&\, \text{false}) \leftrightarrow \text{false}$																																		\\
$\neg{}\mathcal{X} \leftrightarrow (\mathcal{X} \leftarrow \text{false})$																																\\
$\neg\text{false} \leftrightarrow \text{true}$
%\end{eqnarray*}

\end{document}