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%%% file:      s4.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2018/11/22                                                    %%%
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\begin{document}

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\title{4. domáca úloha z predmetu 1-AIN-121 Matematika (1) ZS 2018/19}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Nech $f_n$ je postupnosť prirodzených čísel definovaná vzťahom:
\begin{align*}
f_0 & = 0
\\
f_1 & = 1
\\
f_2 & = 2
\\
f_{n+3} & = f_n - 2 f_{n+2} + 3
.
\end{align*}
Uvažujme ďalej funkciu $g$ definovanú primitívnou rekurziou so substitúciou
v parametroch:
\begin{align*}
g(0,a,b,c) & = c
\\
g(n+1,a,b,c) & = g(n,b,c,a-2c+3)
.
\end{align*}
Dokážte, že pre každé prirodzené číslo $n$ platí
\begin{align*}
f_{n+2} = g(n,0,1,2)
.
\end{align*}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech $A,B,C$ sú podmnožiny základnej množiny $U$ také, že
\begin{gather}
A \subseteq B \rightarrow A \cap B \nsubseteq C
\label{ax1}
\\
A \subseteq B \lor A \subseteq C
\label{ax2}
\\
A \cap C \subseteq B
.
\label{ax3}
\end{gather}
Dokážte, že potom platí:
\begin{gather}
A \cap B \nsubseteq C
.
\label{thm}
\end{gather}
\emph{Návod.}
V dôkaze doporučujeme využiť túto vlastnosť
\begin{align}
X \subseteq Y \leftrightarrow X \cap Y = X
.
\label{lema}
\end{align}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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