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%%% file:      s4.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2018/11/22                                                    %%%
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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{\nA}{\ensuremath{\mathcal A}}

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\begin{document}

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\title{4. 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}

Nech $X \subseteq A$ a $Y \subseteq B$ sú konečné množiny. 
Nech $|X| = k$, $|A| = m$, $|Y | = l$ a $|B| = n$.
Koľko je zobrazení $f$ z $A$ do $B$ takých, že $f(X) \supseteq Y$?

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech
\(
A = \{ x \in \nN \mid \exists y (x = 2y \lor x = y^2) \}
\).
Zostrojte surjektívne zobrazenie $f : \nN \to A$.
Tým dokážete, že $A$ je spočítateľná množina.

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Nájdite explicitné vyjadrenie pre tieto (nekonečné) postupnosti:
\begin{enumerate}
\item $1,1,-1-1,1,1,-1-1,\ldots$,
\item $0,-1,1,-2,2,-3,3,\ldots$.
\end{enumerate}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

Dokážte, že univerzum všetkých nekonečných dyadických slov:
\begin{align*}
\{1,2\}^{\nN} = \bigl\{ f \bigm| f : \nN \to \{1,2\} \bigr\}
\end{align*}
je nespočítateľná množina.

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

Ukážte, že množina všetkých nekonečných postupností znakov nad abecedou $\nA$, 
$|\nA| \geq 2$, je nespočítateľná.

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}

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