%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%                                                                          %%%
%%% file:      s4.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2016/11/21                                                    %%%
%%%                                                                          %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[12pt,a4paper]{article}

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

\usepackage{amsmath}
\usepackage{amssymb}

\usepackage{lmodern}
\usepackage{indentfirst}

\newcommand{\nN}{\ensuremath{\mathbb{N}}}
\newcommand{\aT}[1]{\ensuremath{\mathrm{T}_{#1}}}
\newcommand{\aIt}[1]{\ensuremath{\mathrm{T}_{#1}^{-1}}}

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

\begin{document}

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

\title{4. domáca úloha z predmetu 1-AIN-160 Matematika (3) ZS 2016/17}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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

\section*{1. príklad}

Dokážte, že zobrazenie $f: A \to B$ je bijektívne práve vtedy, keď existuje 
zobrazenie $g: B \to A$ také, že
\[
\forall x {\in} A\, g(f(x)) = x \land \forall y {\in} B\, f(g(y)) = y
.
\]

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

\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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

\section*{2. príklad}

Nech
\(
A = \{ (x,y) \in \nN^2 \mid x \leq y \}
\).
Zostrojte bijekciu $f: A \to \nN$ a zobrazenia $g,h: \nN \to A$ také, že
\[
\forall x,y {\in} A\, g(f(x,y)) = x \land \forall x,y {\in} A\, h(f(x,y)) = y
.
\]
\emph{Návod.}
Bijekciu $f$ navrhnite tak, ako je to uvedené v tabuľke: 
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
$f(x,y)$ & $0$ & $1$ & $2$ & $3$ & $4$  & $5$  & $6$  \\ \hline\hline
$0$      & $0$ & $1$ & $3$ & $6$ & $10$ & $15$ & $21$ \\ \hline
$1$      &     & $2$ & $4$ & $7$ & $11$ & $16$ & $22$ \\ \hline
$2$      &     &     & $5$ & $8$ & $12$ & $17$ & $23$ \\ \hline
$3$      &     &     &     & $9$ & $13$ & $18$ & $24$ \\ \hline
$4$      &     &     &     &     & $14$ & $19$ & $25$ \\ \hline
$5$      &     &     &     &     &      & $20$ & $26$ \\ \hline
$6$      &     &     &     &     &      &      & $27$ \\ \hline
\end{tabular}
\end{center}
Riešenie vyjadrite pomocou \aT{n} a \aIt{m}.

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

\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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

\section*{3. príklad}

Dokážte, že množina všetkých zobrazení z množiny celých čísel do množiny celých
čísel je nespočítateľná množina.

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

\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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

\end{document}

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