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%%% file:      s5.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2016/11/23                                                    %%%
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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{\nR}{\ensuremath{\mathbb R}}
\newcommand{\nZp}{\ensuremath{\mathbb Z}^+}
\newcommand{\nU}{\ensuremath{\mathcal U}}
\newcommand{\aP}[1]{\ensuremath{\mathcal{P}({#1})}}
\newcommand{\rDivides}{\ensuremath{\mathrel{|}}}

\begin{document}

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\title{5. domáca úloha z predmetu \\ 1-AIN-121 Matematika (1) ZS 2016/17}

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Ukážte, že pre ľubovoľnú univerzálnu množinu $\nU$ a ľubovoľné
\(
A, B \subseteq \nU
\)
sú nasledujúce výroky ekvivalentné:
\begin{enumerate}
\item $A \subseteq B$,
\item $\overline{A} \cup B = \nU$,
\item $\overline{B} \subseteq \overline{A}$.
\end{enumerate}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Pre množiny $A, B \subseteq \nU$ dokážte: 
\begin{enumerate}
\item $\aP{A \cap B} = \aP{A} \cap \aP{B}$.
\item $\aP{A} \cup \aP{B} \subseteq \aP{A \cup B}$.
\item 
Ukážte na príklade, že rovnosť
\(
\aP{A \cup B} = \aP{A} \cup \aP{B}
\)
nemusí vždy platiť.
\end{enumerate}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

\begin{enumerate}
%
\item 
Nech $\nU = \nR$ a nech $I = \nZp$. 
Pre každé $n \in \nZp$ nech $A_n = [-2n, 3n]$.
Nájdite každé z nasledujúcich:
\begin{enumerate}
\item $A_3$,
\item $A_3 \setminus A_4$,
\item $A_3 \mathbin{\triangle} A_4$, 
\item $\displaystyle \bigcap_{n = 1}^7 A_n$,
\item $\displaystyle \bigcup_{n \in \nZp} A_n$,
\item $\displaystyle \bigcap_{n = 1}^\infty A_n$.
\end{enumerate}
Tu $\triangle$ označuje symetrický rozdiel množín.
%
\item 
Použite definíciu zovšeobecneného zjednotenia a dokážte nasledujúce
zovšeobecnenie distributívneho zákona:
\[
A \cap (\bigcup_{k \in I} B_k) = \bigcup_{k \in I} (A \cap B_k).
\]
Tu $I$ je množina indexov.
%
\end{enumerate}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

Použitím  matematickej indukcie dokážte:
\begin{enumerate}
%
\item 
Súčet prvých $n$ nepárnych kladných celých čísel je $n^2$, t. j. 
\[
\sum_{i = 1}^n (2i-1) = n^2.
\]
%
\item 
Pre všetky $n > 3$ platí $2^n < n!$.
%
\item 
Pre $n \in \nZp$, $5 \rDivides (n^5 - n)$.

\emph{Poznámka.} Symbol $a \rDivides b$ znamená, že celé číslo $a$ celočíselne 
delí celé číslo $b$. Ekvivalentne povedané: $b = k \cdot a$ pre nejaké celé 
číslo $k$.
%
\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
Na prednáške sme dvoma rôznymi spôsobmi dokázali tvrdemie: 
\begin{quote}
Ak $A$ je $n$-prvková množina, tak $|\aP{A}| = 2^n$.
\end{quote}
Teraz to dokážte ešte tretím spôsobom - s použitím matematickej indukcie. 
\item 
Dokážte pomocou indukcie: Ak $n$ ľudí stojí v rade a ak osoba na začiatku radu
je žena a osoba na konci radu je muž, tak niekde v rade stoja za sebou žena a
hneď za ňou muž.
\end{enumerate}

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}
