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%%% file:      s4.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2019/11/18                                                    %%%
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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}}

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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 $f : \nZ \to \nN$ je definovaná takto:
\[
f(x) = 
\left\{
\begin{array}{cl}
2x-1 & \text{ak $x > 0$,}
\\
-2x & \text{ak $x \leq 0$.}
\end{array}
\right.
\]
\begin{enumerate}
\item 
Dokážte, že $f$ je injektívna a surjektívna.
\item 
Nájdite $f^{-1}$.
\end{enumerate}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Dokážte, že 
\begin{enumerate}
\item
množina všetkých (konečných) slov nad konečnou abecedou je spočítateľná množina.
\item 
systém všetkých konečných podmnožín množiny prirodzených čísel je spočítateľná 
množina.
\end{enumerate}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

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

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. 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 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

Dokážte: 
Ak $A, B$ sú spočítateľné množiny, potom aj ich karteziánsky súčin $A \times B$ 
je spočítateľná množina.

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}

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