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%%% file:      s3.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2014/04/10                                                    %%%
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\begin{document}

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\title{3. domáca úloha z predmetu 1-AIN-160 Diskrétna matematika (2) LS 2013/14}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Nech $R$ a $S$ sú binárne relácie na množine $A$.
Nech ďalej $X \subseteq A$.
Dokážte, že platí:
%%%
\begin{gather*}
(R \cup S)[X] = R[X] \cup S[X]
.
\end{gather*}
%%%

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech $X \subseteq A$ a $Y \subseteq B$ sú konečné množiny.
Nech
%%%
\begin{align*}
|X| = k \quad |A| = m \quad |Y| = l \quad |B| = n
.
\end{align*}
%%%
Koľko je všade definovaných relácií $R$ z $A$ do $B$ takých, že $R[X] \supseteq Y$?

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Nech $X \subseteq A$ a $Y \subseteq B$ sú konečné množiny.
Nech
%%%
\begin{align*}
|X| = k \quad |A| = m \quad |Y| = l \quad |B| = n
.
\end{align*}
%%%
Koľko je jednoznačných relácií $R$ z $A$ do $B$ takých, že $R[X] = Y$?

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\end{document}

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