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%%% file:      s3.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2018/11/03                                                    %%%
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\begin{document}

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\title{3. 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}

Nájdite obor pravdivosti výrokovej formy
\begin{align*}
5 \leq 4|x-1| + |2-3x|
\end{align*}
definovanej nad oborom reálnych čísel. Dokážte, že vaše riešenie je správne.

\emph{Návod.}
V dôkaze doporučujeme využiť niektorú z nasledujúcich vlastností absolútnej 
hodnoty reálneho čísla:
\begin{gather}
\forall x \forall y\, |xy| = |x||y|
\label{times}
\\
\forall x \forall y (y \leq |x| \leftrightarrow y \leq x \lor y \leq -x)
\label{lb}
\\
\forall x \forall y (|x| \leq y \leftrightarrow x \leq y \land -x \leq y)
.
\label{ub}
\end{gather}


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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech $a_n$ je postupnosť čísel definovaná vzťahom:
\begin{align*}
a_0 & = -1
\\
a_1 & = 3
\\
a_{n+2} & = 6 a_n - a_{n+1} + 4n + 1
.
\end{align*}
Dokážte, že pre každé prirodzené číslo $n$ platí rovnosť
\begin{align*}
a_n = 2^n - (-3)^n - n - 1
.
\end{align*}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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