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%%% file:      s5.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
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%%% version:   2021/11/28                                                    %%%
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\begin{document}

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\title{5. domáca úloha z predmetu 1-AIN-121 Matematika (1) ZS 2021/22}

\author{Ján Komara}

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\date{\today}

\maketitle

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\section*{1. príklad}

Nech $R \subseteq A \times B$, $X \subseteq A$ a $Y \subseteq B$.
Dokážte, že platí
\begin{align*}
R^{-1}[Y] \subseteq X \leftrightarrow R[\overline{X}] \subseteq \overline{Y}
.
\end{align*}

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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 také, 
že $|X| = k$, $|A| = m$, $|Y| = l$ a $|B| = n$.
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 také, 
že $|X| = k$, $|A| = m$, $|Y| = l$ a $|B| = n$.
Koľko je zobrazení $f$ z $A$ do $B$ takých, že $f(X) = Y$?

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\end{document}

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