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%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2018/10/07                                                    %%%
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\documentclass[12pt,a4paper]{article}

\usepackage[slovak]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{amsmath}
\usepackage{amssymb}

\usepackage{lmodern}
\usepackage{indentfirst}

\renewcommand{\labelenumi}{{\normalfont (\alph{enumi})}}
\renewcommand{\labelenumii}{{\normalfont \roman{enumii}.}}

\newcommand{\nN}{\ensuremath{\mathbb N}}
\newcommand{\nZ}{\ensuremath{\mathbb Z}}
\newcommand{\nR}{\ensuremath{\mathcal R}}

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\begin{document}

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\title{1. domáca úloha z predmetu \\ 1-AIN-160 Matematika (3) ZS 2018/19}

\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 $\nN \times \nN$?
Zistite či niektoré z týchto relácií sú ekvivalencie.
\begin{enumerate}
\item 
$(x,y) \in \nR \Leftrightarrow x + y$ je párne číslo.
\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 $A = \{w,x,y,z\}$. Určite počet relácií na $A$, ktoré sú
\begin{enumerate}
\item 
reflexívne a symetrické;
\item 
symetrické a obsahujú $(x,y)$;
\item 
antisymetrické a obsahujú $(x,y)$;
\item 
symetrické a antisymetrické.
\end{enumerate}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Nech $A = \{1,2,3,4,5,6,7\}$.
Koľko symetrických relácií na $A$ obsahuje presne
\begin{enumerate}
\item 
päť usporiadaných dvojíc?
\item 
osem usporiadaných dvojíc?
\end{enumerate}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

\begin{enumerate}
\item 
Nech $\nR_1$ a $\nR_2$ sú relácie na $A$. Dokážte alebo vyvráťte:
\begin{enumerate}
\item 
Ak $\nR_1$ a $\nR_2$ sú reflexívne (symetrická, antisymetrická, tranzitívna), 
tak aj  $\nR_1 \cap \nR_2$ je reflexívna (symetrická, antisymetrická, 
tranzitívna). 
\emph{Poznámka.}
Tu si z uvedených vlastností vyberte ľubovolné dve a príklad spravte pre ne.
\item 
Ak $\nR_1$ a $\nR_2$ sú reflexívne (symetrická, antisymetrická, tranzitívna), 
tak aj  $\nR_1 \cup \nR_2$ je reflexívna (symetrická, antisymetrická, 
tranzitívna). 
\emph{Poznámka.}
Tu si z uvedených vlastností vyberte zvyš\-né dve, tie ktoré ste si nevybrali 
v časti (a), a príklad spravte pre ne.
\end{enumerate}
\item
Pre každé z nasledujúcich tvrdení o reláciách na množine $A$, kde $|A| = n$, 
určite, či je pravdivé alebo nepravdivé. Ak je nepravdivé, uveďte kontrapríklad.
\begin{enumerate}
\item 
Ak $\nR$ je relácia na $A$ a $|\nR| \geq n$, tak $\nR$ je reflexívna.
\item 
Ak $\nR$ je ekvivalencia na $A$, tak  $n \leq |\nR| \leq n^2$. 
\end{enumerate}
\end{enumerate}

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

\begin{enumerate}
\item 
Nech $n \in \nZ^+$, $n>1$ a nech $A$ je množina všetkých kladných celočíselných 
deliteľov čísla $n$. Definujme reláciu $\nR$ na $A$ takto:
$(x,y) \in  \nR $, ak $x$ delí $y$. 
Určite koľko usporiadaných dvojíc obsahuje relácia $\nR$ 
pre $n = 20$, $n = 210$ a $n = 13860$.
\item 
Nech $p_1, p_2, p_3$ sú rôzne prvočísla. 
Nech $n, k \in \nZ^+$ také, že $n = p_1^5 p_2^3 p_3^k$. Nech $A$ je množina 
kladných celočíselných deliteľov $n$. Reláciu $\nR$ na $A$ definujeme rovnako 
ako v predchádzajúcej časti: $x \nR y$, ak $x$ delí $y$. 
Nájdite $k$ a $|A|$, ak vieme, že $\nR$ má $5880$ usporiadaných dvojíc.
\end{enumerate}


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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\section*{Bonusový príklad \normalfont{(extra 2 body)}}

Pre reláciu $\nR$ na množine $A$ definujme 
$\nR^0 = I_A = \{(a,a) \mid a \in A\}$. 
Ak $|A|= n$, dokážte, že existujú $s,t \in \nN$, 
kde $0 \leq s < t \leq 2^{n^2}$, pre ktoré  $\nR^s  = \nR^t$.  

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\subsection*{Riešenie bonusového príkladu}

Táto časť obsahuje riešenie bonusového príkladu.

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\end{document}

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