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%%% file:      s2.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2017/10/25                                                    %%%
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\documentclass[12pt,a4paper]{article}

\usepackage[slovak]{babel}
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\usepackage{lmodern}
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\renewcommand{\labelenumi}{{\normalfont (\alph{enumi})}}
\renewcommand{\labelenumii}{{\normalfont (\roman{enumi})}}

\newcommand{\nN}{\ensuremath{\mathbb N}}
\newcommand{\nR}{\ensuremath{\mathcal R}}
\newcommand{\aP}[1]{\ensuremath{{\mathcal P}(#1)}}

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\begin{document}

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\title{2. domáca úloha z predmetu \\ 1-AIN-160 Matematika (3) ZS 2017/18}

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Ktoré vlastnosti (z tých, čo sme definovali na prednáške) má a ktoré nemá
relácia \nR\ definovaná na $\nN \times \nN$?
\begin{enumerate}
\item 
$(x,y) \in \nR \Leftrightarrow x + y$ je párne číslo.
\item 
$(x,y) \in \nR \Leftrightarrow x \mid y$.
\item 
$(x,y) \in \nR \Leftrightarrow x + y < 100$.
\item 
$(x,y) \in \nR \Leftrightarrow |x - y| \leq 1$.
\end{enumerate}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech $A = \{u,v,w,x,y,z\}$. Určite počet relácií na $A$, ktoré sú
\begin{enumerate}
\item 
reflexívne a symetrické (zároveň);
\item 
ekvivalencie
\item 
reflexívne a symetrické, ale nie sú tranzitívne;
\item 
ekvivalencie s práve dvoma triedami rozkladu;
\item 
ekvivalencie s $w \in [x]$;
\item 
ekvivalencie s $w \in [x]$ a $y \in [z]$;
\item 
ekvivalencie s $w \in [x]$, $y \in [z]$ a $[x] \ne [z]$.
\end{enumerate}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Nech $A = \{1,2,3,4,5\} \times \{1,2,3,4,5\}$. Definujme na $A$ reláciu \nR\
nasledovne: $(x_1,y_1) \nR (x_2,y_2)$, ak $x_1 + y_1 = x_2 + y_2$.
\begin{enumerate}
\item 
Overte, že \nR\ je ekvivalencia.
\item 
Určite triedy ekvivalencie $[(1,4)]$, $[(4,2)]$ a $[(1,1)]$.
\item 
Určite rozklad množiny $A$ indukovaný reláciou \nR.
\end{enumerate}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. 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 zobrazení $f$ z $A$ do $B$ takých, že $f(X) \supseteq Y$?

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

Nech $A$ je neprázdna množina a nech $B$ je jej jedna fixovaná podmnožina.
Definujme reláciu \nR\ na \aP{A} takto: $X \nR Y$ pre $X,Y \subseteq A$, ak
$B \cap X = B \cap Y$.
\begin{enumerate}
\item 
Overte, že \nR\ je ekvivalencia na \aP{A}.
\item 
Pre $A = \{1,2,3\}$ a $B = \{1,2\}$ nájdite rozklad \aP{A} indukovaný \nR.
\item  
Pre $A = \{1,2,3,4,5\}$ a $B = \{1,2,3\}$ nájdite $[X]$, ak $X = \{1,3,5\}$.
\item  
Koľko tried rozkladu bude v rozklade indukovanom \nR\ pre $A = \{1,2,3,4,5\}$ a 
$B = \{1,2,3\}$?
\end{enumerate}

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}

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