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%TCIDATA{Created=Tue Nov 22 22:18:52 2011}
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\begin{center}
{\large Matematikai logika}

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\ H\'{a}zi feladat, 2011.11.22.
\end{center}

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1. Fogalmazza meg pontosan azt a defin\'{i}ci\'{o}t, hogy az adott $f(n,x)$ 
\newline
f\"{u}gv\'{e}nysorozat egyenletesen konverg\'{a}l az adott $F(x)$ -hez a sz%
\'{a}megyenesen

($\varepsilon >0$ -ra, valamely\textit{\ term\'{e}szetes} $N$-re, ha $n>N$,
akkor $\left| f(n,x)-F(x)\right| <\varepsilon $), majd formaliz\'{a}lja a k%
\"{o}vetkez\H{o} $\mathcal{L}$ els\H{o}rend\H{u} nyelven. $\mathcal{L}$: $%
f,F,-,abs,0;<,>,T$

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2. a) Hozza valamely prenex norm\'{a}l form\'{a}ra a k\"{o}vetkez\H{o} formul%
\'{a}t: 
\begin{equation*}
\forall x(\forall yRxy\rightarrow \forall y\rceil (Qxy\rightarrow Lxy))
\end{equation*}

\ \ \ b) Az el\H{o}z\H{o} prenex form\'{a}t hozza er\H{o}s Skolem alakra 
\'{e}s adja meg az \'{i}gy nyert kl\'{o}zokat.

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3. Igazolja analitikus fa seg\'{i}ts\'{e}g\'{e}vel, hogy
\begin{equation*}
\nvDash \forall x(Px\rightarrow \exists yTy)\wedge \exists xPx\rightarrow
\exists x(Px\wedge Tx)
\end{equation*}

$\qquad $

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4. Formaliz\'{a}lja az adott nyelven az al\'{a}bbi k\"{o}vetkeztet\'{e}st 
\'{e}s vizsg\'{a}lja meg analitikus f\'{a}val, hogy helyes-e?

\textit{Esni fog}. Egy \textit{r\'{e}szeg} bele fog \textit{zuhanni} a
csatorn\'{a}ba. Ha valaki bezuhan a csatorn\'{a}ba, akkor \textit{vizes
lesz. Toni} belezuhant a csatorn\'{a}ba.Teh\'{a}t ha esik, akkor valaki
vizes lesz.$\;\mathcal{L}$:\ $E,Rx,Zx,Vx,T$

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5. a) Ellentmond\'{a}stalan-e a k\"{o}vetkez\H{o} axioma rendszer?
\begin{equation*}
\left\{ \forall x(Px\rightarrow \exists yQxy),\;\exists y\rceil
Qay,\;Pa\right\} 
\end{equation*}

\bigskip b) Mutassa meg, hogy a csoportok axioma rendszere nem komplett (az
axi\'{o}m\'{a}k: asszociativit\'{a}s, 0-elem \'{e}s inverz l\'{e}tez\'{e}se).

\'{A}ll\'{i}t\'{a}sait igazolja!

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