%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%% file:      s3.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2018/11/02                                                    %%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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{\nZ}{\ensuremath{\mathbb Z}}
\newcommand{\nR}{\ensuremath{\mathcal R}}
\newcommand{\nL}{\ensuremath{\mathcal L}}
\newcommand{\nP}{\ensuremath{\mathcal P}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{1. príklad}

Nech $A = \{1,2,3,4,5,6,7\}$ a nech $\nR$ je relácia \emph{byť podmnožinou} 
$\subseteq_A$ definovaná na potenčnej množine $\nP(A)$. 
Už vieme, že táto relácia je čiastočné usporiadanie na $\nP(A)$. 
Nech $M = \{\{1\},\{2\},\{2,3\}\}$.
Nájdite:
\begin{enumerate}
%
\item 
Počet horných ohraničení $M$, ktoré obsahujú
\begin{enumerate}
\item 
štyri prvky,
\item 
päť prvkov.
\end{enumerate}
%
\item 
Počet všetkých horných ohraničení $M$ a počet všetkých dolných ohraničení $M$.
%
\item $\sup(M)$ a $\inf(M)$.
%
\item 
Pripomeňme:
\begin{quote}
\emph{Definícia.}
Čiastočne usporiadaná množina $(A, \nR)$ sa nazýva zväz, ak  $\sup(\{x,y\})$ aj
$\inf(\{x,y\})$ existujú v $A$ pre každé $x,y \in A$.
\end{quote}
Je ${(\cal P}(A),{\subseteq_A})$ zväz? (Svoju odpoveď zdôvodnite.)
%
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{2. príklad}

Dokážte, že čiastočne usporiadaná množina $(A,\nR)$ môže mať nanajvýš jeden 
najmenší prvok.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{3. príklad}

Profesor Grimaldi sa chystá na svoju prezentáciu na seminári z Diskrétnej 
matematiky. Ráno sa oblieka a množina $O$ obsahuje kusy šatstva, ktoré si dá na 
seba, aby pri prezentácii vyzeral slušne:
\begin{align*}
O = 
\{&
\text{opasok}, 
\text{sako}, 
\text{nohavice}, 
\text{košeľa}, 
\text{topánky}, 
\text{ponožky}, 
\text{spodky}, 
\text{tielko},
\\ &
\text{hodinky}, 
\text{kravata}
\}
.
\end{align*}
Na množine $O$ definujme reláciu čiastočného usporiadania $\nR$ nasledovne:
\begin{quote}
$o_1 \nR o_2$, ak oblečenie $o_1$ si musí  profesor obliecť predtým ako si
oblečie oblečenie $o_2$.
\end{quote}
Napríklad, dvojica 
\(
(\text{košeľa}, \text{kravata}) \in \nR
\), 
lebo kravatu si nemôže uviazať bez toho aby mal na sebe košeľu, ale dvojice 
\(
(\text{košeľa}, \text{ponožky}) \notin \nR
\), 
ani 
\(
(\text{ponožky}, \text{košeľa}) \notin \nR
\),
lebo oblečenie košele a ponožiek je nezávislé a môže to urobiť v akomkoľvek 
poradí.
\begin{enumerate}
\item 
Nakreslite Hasseho diagram čiastočne usporiadanej množiny $(O,\nR)$.
\item 
$(O,\nR)$ nie je lineárne usporiadaná. Prečo?
\item 
Nájdite dve rôzne lineárne usporiadania $\nL_1$ a $\nL_2$ množiny $O$, ktoré 
rešpektujú čiastočné usporiadanie $\nR$ (t. j. $ \nR \subseteq \nL_i$ ) a v 
ktorých si profesor Grimaldi nasadí hodinky ako posledné.
\item 
Koľko je všetkých možných lineárnych usporiadaní $\nL$ množiny $O$, ktoré 
rešpektujú čiastočné usporiadanie $\nR$ a v ktorých si profesor Grimaldi 
oblečie ako prvé spodky a nasadí hodinky ako posledné.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{4. príklad}

Nech $S$ je množina, ktorá obsahuje $0$ a všetky kladné celé čísla, ktoré sa 
dajú vytvoriť ako súčty čísel $7$ a $11$.
\begin{quote}
Tak napríklad okrem $0,7$ a $11$, aj $14 \in S$, $18 \in S$,  $21 \in S$ a 
$22 \in S$. $S$ je samozrejme nekonečná množina.
\end{quote}
Na tejto množine $S$ definujme reláciu $\leq_S$ nasledovne:
\begin{quote}
Pre $x,y \in S$, $x \leq_S y$, ak $y-x \in S$.
\end{quote}

\begin{enumerate}
\item 
Ukážte, že takto definovaná relácia $\leq_S$ je čiastočné usporiadanie na $S$.
\item 
Načrtnite ``kúsok''  Hasseho diagramu pre $(S,\leq_S)$.
\item 
Je toto čiastočné usporiadanie lineárne? 
(Svoju odpoveď - ako ostatne máte robiť vždy - zdôvodnite.)
\item 
Nech $M = \{21,25,29\}$. Nájdite $\sup(M)$ a $\inf(M)$, ak existujú.
\item 
Je $(S,\leq_S)$ zväz?
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{5. príklad}

\begin{enumerate}
\item 
Nech $(\nZ^+,\leq_1)$ je množina prirodzených čísel s obvyklým usporiadaním a 
nech $(\nZ^+,\leq _2)$ je množina s usporiadaním deliteľnosti ($a \leq _2 b$, 
ak $a$ delí $b$).
Ukážte, že $(\nZ^+,\leq_1)$ nie je izomorfná s $(\nZ^+,\leq_2)$.
\item
Nech $M$ je množina. Nech $\leq_1$ a $\leq_2$ sú dve rôzne usporiadania množiny 
$M$. 
Ukážte, že ak $(M,\leq_1)$ obsahuje najväčší prvok a $(M,\leq_2)$ neobsahuje
najväčší prvok, tak $(M,\leq_1)$  a $(M,\leq_2)$ nie sú izomorfné.
\item 
Použítím Hasseho diagramov identifikujte všetky izomorfné triedy ČUM (čiastočne 
usporiadaných množín) s piatimi prvkami, ktoré majú v Hasseho diagrame štyri 
úrovne.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
