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%%% file:      s45.tex                                                       %%%
%%% author:    Tatiana Jajcayová, Ján Komara                                 %%%
%%% email:     Jan.Komara@fmph.uniba.sk                                      %%%
%%% copyright: public domain                                                 %%%
%%% version:   2015/11/12                                                    %%%
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\begin{document}

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\title{4. a 5. 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}

Basketbalové družstvo má úspešnosť zhruba 1 výhru z každých troch zápasov 
(t.j. pravdepodobnosť ich výhry v zápase je $\frac{1}{3}$). 
V turnaji hrajú presne tri zápasy.
\begin{enumerate}
\item 
Aká je pravdepodobnosť, že ani raz nevyhrajú?
\item 
Aká je pravdepodobnosť, že vyhrajú tri krát?
\item 
Akú by museli mať úspešnosť, aby ich šanca troch výhier bola viac ako 40\%?
\end{enumerate}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Očkovanie proti kiahňam zaručí úplnú imunitu proti nákaze kiahňami v 95\%
prípadov. Predpokladajme tiež, že ak očkovanie nezafunguje a nevytvorí u
očkovaného úplnú imunitu, tak nemá žiadny efekt - pravdepodobnosť nákazy
kiahňami u človeka, kde očkovanie nezabralo, je presne rovnaká ako u
neočkovaného človeka. Ak je v populácii 30\% zaočkovaných ľudí, aká je
pravdepodobnosť, že človek, ktorý dostal kiahne bol zaočkovaný?

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Test na lupus dáva (správne) pozitívne výsledky v 95\% výsledkov, keď sa chroba
naozaj vyskytuje a dáva (nesprávne) pozitívne výsledky v 10\% výsledkov, keď sa
chroba nevyskytuje. Výskyt lupusu v populácii sa odhaduje na 2\%.
\begin{enumerate}
\item 
Ak má náhodne vybratý človek pozitívny test, aká je pravdepodobnosť, že má lupus?
\item 
Ak má náhodne vybratý človek negatívny test, aká je pravdepodobnosť, že má lupus?
\end{enumerate}


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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\section*{4. príklad}

Predpokladajme, že v klobúku sú tri karty. Jedna z nich je červená na obidvoch
stranách, jedna z nich je čierna na obidvoch stranách a tretia má jednu stranu
červenú a druhú čiernu. Z klobúka náhodne vytiahneme jednu kartu, a vidíme, že
jedna jej strana je červená. Aká je pravdepodobnosť, že aj druhá strana je
červená?

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\subsection*{Riešenie 4. príkladu}

Táto časť obsahuje riešenie 4. príkladu.

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\section*{5. príklad}

Uvažujme zložený výrok: $( p \land q) \lor (p \lor \lnot q)$.
\begin{enumerate}
\item 
Nájdite dve slovenské vety $v_1$ a $v_2$ tak, aby po substitúcii vety $v_1$ za
$p$ a substitúcii vety $v_2$ za $q$ vznikol pravdivý výrok.
\item 
Nájdite dve slovenské vety $v_1$ a $v_2$ tak, aby po substitúcii vety $v_1$ za
$p$ a substitúcii vety $v_2$ za $q$ vznikol nepravdivý výrok.
\end{enumerate}

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\subsection*{Riešenie 5. príkladu}

Táto časť obsahuje riešenie 5. príkladu.

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\section*{6. príklad}

Zjednodušte:
\[
\lnot [ \lnot [(p \lor q) \land r] \lor \lnot q]
.
\]

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\subsection*{Riešenie 6. príkladu}

Táto časť obsahuje riešenie 6. príkladu.

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\section*{7. príklad}

Nájdite, ak existujú, \emph{všetky} pravdivostné ohodnotenia atomických výrokov
$p,q,r,s,t$, pre ktoré je nasledujúci výrok nepravdivý:
\[
[(p \land q) \land r] \rightarrow (r \lor t)
.
\]

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\subsection*{Riešenie 7. príkladu}

Táto časť obsahuje riešenie 7. príkladu.

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\section*{8. príklad}

Uvažujme nasledujúcu výrokovú funkciu:
\[ 
p(x,y):\ y - x = y + x^2 
,
\]
kde univerzum každej premennej je množina celých čísel.

Určite pravdivostnú hodnotu každého z nasledujúcich výrokov:
\begin{enumerate}
\item $p(0,0)$,
\item $p(1,1)$,
\item $p(0,1)$,
\item $\forall y\; p(0,y)$,
\item $\exists y\; p(1,y)$,
\item $\forall x \exists y\; p(x,y)$,
\item $\exists y \forall x\; p(x,y)$,
\item $\forall y \exists x\; p(x,y)$.
\end{enumerate}

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\subsection*{Riešenie 9. príkladu}

Táto časť obsahuje riešenie 9. príkladu.

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\section*{10. príklad}

Nech $S = \{1,2,3,\ldots,29,30\}$. 
Koľko podmnožín $A$ množiny $S$ spĺňa nasledujúce podmienky:
\begin{enumerate}
\item 
$|A| = 5$?
\item 
$|A| = 5$ a najmenší prvok v $A$ je $5$?
\item 
$|A| = 5$ a najmenší prvok v $A$ je menší ako $5$?
\item 
$A$ obsahuje aspoň jedno párne číslo?
\end{enumerate}

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\subsection*{Riešenie 10. príkladu}

Táto časť obsahuje riešenie 10. príkladu.

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\section*{11. príklad}

Dokážte alebo vyvráťte (tým, že uvediete protipríklad) nasledujúce tvrdenie:
\begin{quote}
Pre ľubovoľné množiny $A,B,C \subseteq \mathcal{U}$ platí: 
ak  $A \subseteq B$ a $B \nsubseteq C$, tak $A \nsubseteq C$.
\end{quote}
(Pozn.: Vennove diagramy sú iba vizualizácia, nie dôkaz.)

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\subsection*{Riešenie 11. príkladu}

Táto časť obsahuje riešenie 11. príkladu.

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\end{document}

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