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%%% file:      s8.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2020/12/05
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\begin{document}

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\title{8. domáca úloha z predmetu 1-AIN-121 Matematika (1) ZS 2020/21}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Dokážte tvrdenie:
\begin{quote}
\emph{
Zobrazenie $f : A \to B$ je injektívne práve vtedy, 
keď pre každé $X_1 \subseteq A$ a $X_2 \subseteq A$ platí rovnosť
\(
f(X_1 \cap X_2) = f(X_1) \cap f(X_2)
\).
}
\end{quote}
\emph{Návod.}
V dôkaze doporučujeme využiť niektorú z nasledujúcich vlastností.
Ich použite netreba explicitne vyznačovať.
\begin{gather}
\forall x \forall y (\{x\} \cap \{y\} = \emptyset \leftrightarrow x \ne y)
\label{eq:single:empty}
\\
\forall A \forall B\, A \cap B = B \cap A
\label{eq:cap:com}
\\
\forall A \forall B\, A \cap B \subseteq A
\label{eq:cap:subset}
\\
\forall A \forall B \forall C 
(C  \subseteq A \cap B \leftrightarrow C \subseteq A \land C \subseteq B)
\label{eq:subset:cap}
\\
\forall A \forall B
(A = B \leftrightarrow A \subseteq B \land B \subseteq A)
\label{eq:equal:subset}
.
\end{gather}
Nech $f : A \to B$. Potom
\begin{gather}
f(\emptyset) = \emptyset
\label{eq:f:obraz:empty}
\\
\forall x \in A\, f(\{x\}) = \{f(x)\}
\label{eq:f:obraz:single}
\\
\forall X_1 \forall X_2
\bigl(X_1 \subseteq X_2 \subseteq A \rightarrow f(X_1) \subseteq f(X_2)\bigr)
\label{eq:f:obraz:mono}
.
\end{gather}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech
\(
A = \{ 1,2,3,4,5,6 \}
\)
a
\(
B = \{ 1,2,3,4,5,6,7,8,9 \}
\).
Koľko je injektívnych zobrazení $f : A \to B$ takých, že
\(
f(\{ 1,2,3,4 \}) \supseteq \{ 1,2 \}
\)?

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\end{document}

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