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%%% file:      v4.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2020/10/26                                                    %%%
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\title{Vzorový príklad pre 4. domácu úlohu}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{Príklad}

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

\emph{Návod.}
V dôkaze využite tieto dve vlastnosti absolútnej hodnoty reálnych čísel:
\begin{gather}
\forall x \forall y (y \leq |x| \leftrightarrow y \leq x \lor y \leq -x)
\label{abs:lb}
\\
\forall x \forall y (|x| \leq y \leftrightarrow x \leq y \land -x \leq y)
.
\label{abs:ub}
\end{gather}

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\subsection*{Riešenie}

Najprv dokážeme, že platí
\begin{align}
\forall x 
\left(|3x| \leq |2x-1| \leftrightarrow -1 \leq x \leq \frac{1}{5}\right)
.
\label{lema}
\end{align}
Nech $x$ je ľubovolné, ale pevne zvolené reálne číslo. Potom
\begin{align*}
&
|3x| \leq |2x-1|
 \stackrel{\eqref{abs:lb}}{\Leftrightarrow}
|3x| \leq 2x-1 \lor |3x| \leq -(2x-1)
 \stackrel{2\times\eqref{abs:ub}}{\Leftrightarrow}
\\ &
3x \leq 2x-1 \land -3x \leq 2x-1 \lor 3x \leq -(2x-1) \land -3x \leq -(2x-1)
 \Leftrightarrow 
\displaybreak[0]
\\ &
x \leq -1 \land 1 \leq 5x \lor 5x \leq 1 \land -1 \leq x
 \Leftrightarrow  
\\ &
\frac{1}{5} \leq x \leq -1 \lor -1 \leq x \leq \frac{1}{5}
 \Leftrightarrow  
-1 \leq x \leq \frac{1}{5}
.
\end{align*}
Tým sme dokázali pomocné tvrdenie \eqref{lema}.

Obor pravdivosti výrokovej formy \eqref{vf} je preto rovný množine
\begin{align*}
\left\{ x \in R \biggm| -1 \leq x \leq \frac{1}{5} \right\}
.
\end{align*}

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