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%%% file:      s6.tex                                                        %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2015/11/29                                                    %%%
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\begin{document}

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\title{6. domáca úloha z predmetu \\ 1-AIN-121 Matematika (1) ZS 2015/16}

\author{Tatiana Jajcayová \and Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Univerzálna množina nech je množina celých čísel a nech $p(x)$, $q(x)$ a $r(x)$
sú definované nasledovne:
\begin{align*}
&
p(x): x > 0
\\ &
q(x): x^2 + 2x - 15 = 0
\\ &
r(x): \text{$x$ je párne.}
\end{align*}
Určte, ktoré z nasledujúcich výrokov sú pravdivé a ktoré nepravdivé. Ak je výrok
pravdivý, vysvetlite preč. Ak je nepravdivý, uveďte protipríklad.
\begin{enumerate}
\item $\forall x \: [q(x) \rightarrow p(x)]$,
\item $\exists x \: [q(x) \rightarrow r(x)]$,
\item $\forall x \: [q(x) \rightarrow \lnot r(x)]$,
\item $\exists x \: [p(x) \land q(x) \land \lnot r(x)]$.
\end{enumerate}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nájdite negáciu nasledujúceho výroku:
\begin{quote}
Ak všetci študenti poctivo študujú, tak všetci pedagógovia sú šťastní.
\end{quote}
Napíšte svoju odpoveď ako slovenskú vetu (bez použitia logických symbolov).
Obor tu tvoria všetci študenti a pedagógovia FMFI.
Poznámka: nestačí povedať \uv{Nie je pravda, že \ldots}.

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Pre množiny $A, B \subseteq \mathcal{U}$ dokážte: 
\begin{enumerate}
\item 
$\mathcal{P}(A \cap B) = \mathcal{P}(A) \cap \mathcal{P}(B)$.
\item 
$\mathcal{P}(A) \cup \mathcal{P}(B) \subseteq \mathcal{P}(A \cup B)$.
\item 
Ukážte na príklade, že 
$\mathcal{P}(A \cup B) = \mathcal{P}(A) \cup \mathcal{P}(B)$ 
nemusí vždy platiť.
\end{enumerate}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

Použitím  matematickej indukcie dokážte:
\begin{enumerate}
\item 
Pre všetky $n > 4$ platí $n^2 < 2^n$. 
\item 
Pre všetky $n > 3$ platí $2^n <  n!$. 
\item
Pre všetky celé kladné čísla $n$ platí $5 \mid (n^5 - n)$.

Poznámka: 
Symbol $a \mid b$ znamená, že celé číslo $a$ delí bezo zvyšku celé číslo $b$.
Ekvivalentne povedané: $b = k \times a$  pre nejaké celé číslo $k$.
\item 
Súčet prvých $n$ nepárnych kladných celých čísel je $n^2$, t. j. dokážte 
$\sum_{i = 1}^{n} (2i-1) = n^2$.
\end{enumerate}

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

\begin{enumerate}
\item 
Dokážte, že pre všetky celé čísla $n$ platí: 
ak 3 delí  $n^2$, tak 3 delí $n$.
\item 
Dokážte, že $\sqrt{3}$ je iracionálne číslo.
\end{enumerate}

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\end{document}

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