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%%% file:      s5.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2017/12/03                                                    %%%
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\begin{document}

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\title{5. domáca úloha z predmetu 1-AIN-121 Matematika (1) ZS 2017/18}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}
\setcounter{equation}{0}

Nech $A,B,C$ sú ľubovolné množiny. Dokážte, že platí:
\begin{align*}
A \cap C = B \cap C \land A \cup C = B \cup C \rightarrow A = B
.
\end{align*}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}
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Nech $A,B,C$ sú podmnožiny základnej množiny $U$ také, že
\begin{gather}
A \subseteq B \rightarrow A \cap B \nsubseteq C
\label{ax1}
\\
A \subseteq B \lor A \subseteq C
\label{ax2}
\\
A \cap C \subseteq B
.
\label{ax3}
\end{gather}
Dokážte, že potom platí:
\begin{gather}
A \cap B \nsubseteq C
.
\label{thm}
\end{gather}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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