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%%% file:      s1.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2014/03/11                                                    %%%
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\begin{document}

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\title{1. 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 $f: A \to B$ a $Y_1,Y_2 \subseteq B$.
Dokážte, že platí:
%%%
\begin{gather*}
f^{-1}(Y_1 \setminus Y_2) = f^{-1}(Y_1) \setminus f^{-1}(Y_2)
.
\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*}
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Koľko je zobrazení $f$ z $A$ do $B$ takých, že $f(X) \supseteq Y$?

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Funkcia
\(
f : \{ 1,\dotsc,m \} \to \{ 1,\dotsc,n \}
\)
je monotónne rastúca, ak
%%%
\begin{align*}
\forall i \forall j
\bigl(
1 \leq i < j \leq m \rightarrow f(i) \leq f(j)
\bigr)
.
\end{align*}
%%%
Nech $1 \leq k \leq m$ a $1 \leq l \leq n$.
Koľko je monotónne rastúcich funkcií z $\{ 1,\dotsc,m \}$ do $\{ 1,\dotsc,n \}$
takých, že $f(k) = l$?

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\end{document}

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