%LaTeX2e
% Fuzzy Logic, transparencies
\documentclass{slides}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exhyphenpenalty=10000
%%%%%%%%%% Èíslované objekty (definice, vìta, dùkaz, pøíklad) %%%%%%%%%%

\def\numero{\global\advance\objnum by1 \the\kapnum.\the\objnum}
\def\dalsiodstavec#1{\removelastskip\medskip\goodbreak
   \noindent{\bf #1. }\barva
   \ifnextchar[{\ulozlabel}{\ignorespaces}}
\def\ulozlabel[#1]{\expandafter\ifx\csname c:#1\endcsname\relax
     \expandafter\edef\csname c:#1\endcsname {\the\kapnum.\the\objnum}%
     \raise15pt\hbox{\aimlink{c:#1}}%
   \else\errmassage{duplikace labelu [#1]}%
   \fi \ignorespaces}
\def\definice{\def\konec{\removelastskip\medskip}\dalsiodstavec{Definice~\numero}}
\def\tabulka{\def\konec{\removelastskip\medskip}\dalsiodstavec{Tabulka~\numero}}

%%%%%%%%%% Matematické zkratky %%%%%%%%%%%%%%%%%
\def\0{{\bf0}}
\def\1{{\bf1}}

\def\mem#1{\mu_{#1}}    % membership function
\def\memb#1{\mu_{#1}}   % membership function (brackets needed)
\let\subs=\subseteq
\def\P{{\cal P}}
\def\kde{:\;}
\def\je{:\enspace}
\def\dop#1{\overline{#1\,}}
\let\AND=\wedge \let\OR=\vee \let\NOT=\neg
\let\IMPL=\Rightarrow  \let\EQ=\Leftrightarrow
\def\carky{\thinmuskip=\thickmuskip}
\def\lint{\langle}
\def\rint{\rangle}
\def\ui{\lint 0,1\rint }
\def\fu{fuzzy množin}
\def\F{{\cal F}}
\def\supp{\mathop{\rm Supp}}
\def\core{\mathop{\rm core}}
\def\R{{\bf R}} \def\N{{\bf N}} \def\Z{{\bf Z}}
\def\indexf#1{\if.#1{\vbox to 2pt{\kern-2pt\hbox{.}\vss}}\else
   {\vbox to0pt{\vss\hbox{$\scriptscriptstyle#1$}\kern1.7pt}}\fi}
\def\indexu#1{\if.#1{\textstyle.}\else
   {\vbox{\hbox{$\scriptscriptstyle#1$}\kern-1pt}}\fi}
\def\negf#1{\mathop{\NOT}\limits_\indexf{#1}}
\def\capf#1{\mathbin{\mathop{\cap}\limits_\indexf{#1}}}
\def\cupf#1{\mathbin{\mathop{\cup}\limits^\indexu{#1}}}
\def\orf#1{\mathbin{\mathop{\OR}\limits^\indexu{#1}}}
\def\andf#1{\mathbin{\mathop{\AND}\limits_\indexf{#1}}}
\def\circf#1{\mathbin{\mathop{\circ}\limits_\indexf{#1}}}
\def\bigwedgedot{\mathop{\vbox{\hbox
   {$\displaystyle\bigwedge_\indexf{.}$}\kern-9pt}}\nolimits}
\def\bigveedot{\mathop{\vbox{\kern-5pt\hbox
   {$\displaystyle\bigvee^\indexu{.}$}}}\nolimits}
\def\bigveeS{\mathop{\vbox{\kern-5pt\hbox
   {$\displaystyle\bigvee^\indexu{S}$}}}\nolimits}
\def\luk{\L ukasiewicz}
\def\implf#1#2{\mathbin
   {\mathop{{\rightarrow}\vphantom{\vee}}%
   \limits^\indexu{#1}_{\indexf{#2}\kern3pt}}}
\def\eqf#1#2{\mathbin
   {\mathop{{\leftrightarrow}\vphantom{\vee}}\limits^\indexu{#1}_\indexf{#2}}}
\let\epsilon=\varepsilon
\def\bodik{{}\par\noindent\hbox to\parindent{\hss$\bullet$\hss\hss}}
\let\begitems=\medskip \let\enditems=\medskip

\def\ctvr #1{\mathbin{% ctvereèek jako binární operace
   \mkern 2mu
   \vbox{\hrule\hbox to#1{\vrule height#1\hss\vrule}\hrule}}
   \mkern 2mu }
\def\sq{\mathchoice {\ctvr{4.3pt}}   {\ctvr{4.3pt}}
                    {\ctvr{2.6pt}} {\ctvr{2pt}}}
\def\incirc#1{\setbox0=\hbox{$\bigcirc$}\dimen0=\wd0
   \mathbin{\box0\kern-\dimen0 \hbox to\dimen0
      {\hss$\if.#1\cdot\else#1\fi$\hss}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

Axioms of Boolean algebras
$$
%  \halign{#:\hfil\quad&\hfil$#$&${}#$\hfil\qquad&\hfil$#$&${}#$\hfil\cr 
  \halign{\hfil$#$&${}#$\hfil\qquad&\hfil$#$&${}#$\hfil\cr 
  \neg{\neg A} &= A, \cr
  A\vee B &= B\vee A, & A\wedge B &= B\wedge A, \cr
  (A\vee B)\vee C &= A\vee(B\vee C), &
  (A\wedge B)\wedge C &= A\wedge(B\wedge C), \cr 
  A\wedge(B\vee C) &= (A\wedge B)\vee(A\wedge C), &
   A\vee(B\wedge C) &= (A\vee B)\wedge(A\vee C), \cr
  A\vee A &= A, & A\wedge A &= A, \cr
  A\vee(A\wedge B) &= A, & A\wedge(A\vee B) &= A, \cr
  A\vee \1 &= \1, & 
   A\wedge\0 &= \0, \cr
  A\vee \0 &= A, & A\wedge \1 &= A, \cr 
  \neg A\vee A &= \1, &
  \neg A\wedge A &= \0, \cr
  \neg(A\vee B) &= \neg A \wedge \neg B, &
   \neg(A\wedge B) &= \neg A \vee \neg B. \cr
}
$$

\newpage
Axioms of MV-algebras
$$
%  \halign{#:\hfil\quad&\hfil$#$&${}#$\hfil\qquad&\hfil$#$&${}#$\hfil\cr 
  \halign{\hfil$#$&${}#$\hfil\cr 
  A\orf L B &= B\orf L A, \cr
  (A\orf L B)\orf L C &= A\orf L(B\orf L C), \cr 
  A\orf L\0 &= A, \cr
  \neg{\neg A} &= A, \cr
  \neg\0&=\1, \cr
  A\orf L \1 &= \1, \cr 
  \neg(\neg A\orf L B)\orf L B &= \neg(\neg B\orf L A)\orf L A. \cr
}
$$


\newpage
Lattice operations in MV-algebras
$$
  \halign{\hfil$#$&${}#$\hfil\qquad\qquad&\hfil$#$&${}#$\hfil\cr 
  A\vee B&=\neg(\neg A\orf L B)\orf L B &
  A\wedge B&=\neg(\neg A\andf L B)\andf L B
 \cr
}
$$

Distributivity laws in MV-algebras
$$
  \halign{\hfil$#$&${}#$\hfil\qquad&\hfil$#$&${}#$\hfil\cr 
  A\vee(B\wedge C) &= (A\vee B)\wedge(A\vee C) &
  A\wedge(B\vee C)&= (A\wedge B)\vee(A\wedge C)
 \cr
  A\orf L(B\wedge C) &= (A\orf L B)\wedge(A\orf L C) &
  A\andf L(B\vee C)&= (A\andf L B)\vee(A\andf L C)
 \cr
  A\orf L(B\vee C) &= (A\orf L B)\vee(A\orf L C) &
  A\andf L(B\wedge C)&= (A\andf L B)\wedge(A\andf L C)
 \cr
}
$$
\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tabulka [zakony-operaci] Zákony Booleových algeber.
%%%%%%%%
$$
  \advance\baselineskip by2pt \lineskiplimit=-6pt
  \halign{#:\hfil\quad&\hfil$#$&${}#$\hfil\qquad&\hfil$#$&${}#$\hfil\cr 
  \inl[involuce]%
  involuce&  \negf.(\negf.\alpha) &= \alpha, \cr
  \inl[komutativita]%
  komutativita&  \alpha\orf.\beta &= \beta\orf.\alpha, & 
    \alpha\andf.\beta &= \beta\andf. \alpha, \cr
  \inl[asociativita]%
  asociativita& 
    (\alpha\orf. \beta)\orf. \gamma &= \alpha\orf.(\beta\orf.\gamma), &
    (\alpha\andf. \beta)\andf.\gamma &= \alpha\andf.(\beta\andf.\gamma), \cr
  \inl[distributivita]%
  distributivita& \alpha\andf.(\beta\orf.\gamma) 
    &= (\alpha\andf. \beta)\orf.(\alpha\andf.\gamma), &
    \alpha\orf.(\beta\andf.\gamma) 
    &= (\alpha\orf. \beta)\andf.(\alpha\orf.\gamma), \cr
  \inl[idempotence]%
  idempotence& \alpha\orf.\alpha &= \alpha,& \alpha\andf.\alpha&= \alpha, \cr
  \inl[absorpce]%
  absorpce& \alpha\orf.(\alpha\andf. \beta) &= \alpha, 
    & \alpha\andf.(\alpha\orf. \beta) &= \alpha, \cr
  absorpce s jednièkou a nulou& \alpha\orf.1 &= 1, & 
    \alpha\andf.0 &= 0, \cr
  \inl[neutrální prvky]%
  neutrální prvky& \alpha\orf.0 &= \alpha, & \alpha\andf.1 &= \alpha, \cr
  \inl[zákon: kontradikce]%
  zákon kontradikce& &&\alpha\andf. \negf.\alpha &= 0,  \cr
  \inl[zákon: vylouèeného tøetího]%
  zákon vylouèeného tøetího& \alpha\orf. \negf. \alpha &= 1, \cr
  \inl[zákony: de Morganovy, de Morganovy zákony]%
  de Morganovy zákony& 
   \negf.(\alpha\orf.\beta) &= \negf.\alpha\andf.\negf.\beta, & 
   \negf.(\alpha\andf. \beta) &= \negf.\alpha \orf. \negf.\beta. \cr
}
$$
\konec

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
