\documentclass{article}
\usepackage{slovak}
\usepackage{amssymb}

\newcommand{\RR}{\ensuremath{\mathbb R}}
\newcommand{\VCdim}{\ensuremath{\mathop{\rm VCdim}}}

\begin{document}
\thispagestyle{empty}
\centerline{\LARGE{\bf �lohy k VC-dimenzii}}
\medskip
\centerline{({\bf V�po�tov� u�enie})}

\begin{enumerate}
\item Uk�te, �e ak $X = \RR$ a $H$ je mno�ina v�etk�ch uzavret�ch intervalov, 
tak
$$\Pi_H(m) = 1 + m + \frac{1}{2}m(m-1)$$
\item Pop�te explicitne hypot�zov� priestor $P_1$ a uk�te, �e $\VCdim(P_1) = 2$.
\item Uk�te, �e ak $H$ je hypot�zov� priestor re�lneho perceptr�nu $P_2$, tak
$$\Pi_H(4) = 14$$
\item Nech $H$ m� kone�n� VC dimenziu. Pre $h \in H$ definujme $\bar{h}$:
$$\bar{h} = 1 \Leftrightarrow h(x) = 0, \qquad h: \qquad \to \{0,1\}$$
a nech komplement $H$ je priestor $\{\bar h: h \in H\}$.
Dok�te, �e maj� oba priestory rovnak� VC dimenziu.
\item Dok�te
\begin{itemize}
\item $\Phi(d,m) = \Phi(d,m-1) + \Phi(d-1,m-1), d \ge 1, m \ge 2$
\item $\Phi(d,m) \le m^\alpha, m \ge \alpha \ge 1$
\end{itemize}
\item Mono�len je monot�nny, ak neobsahuje �iadne negovan� liter�ly. Dok�te, �e priestor
monot�nnych mono�lenov definovan�ch na $\{0,1\}^n$ m� VC dimenziu pr�ve $n$.
\item Hypot�zov� priestor $H$ je \emph{line�rne usporiadan�}, ak m� aspo� 2 hypot�zy
a ak pre �ubovo�n� dve $h,g\in H$ plat�
\begin{itemize}
\item bu� $h(x) = 1 \Rightarrow g(x) = 1$
\item alebo $g(x) = 1 \Rightarrow h(x) = 1$.
\end{itemize}
Dok�te, �e ak $H$ je line�rne usporiadan�, tak $\VCdim(H) = 1$.
\item Nech $G_n$ je mno�ina hypot�z z $P_n$, pre ktor� nulov� vektor $\bf o$ je
negat�vny pr�klad. Predpokladajme, �e vzorka $x = (x_1,\dots,x_n)$ je rozbit�
pomocou $G_n$. Pre�o sa �iadne $x_i$ nesmie rovna� $\bf o$? Dok�te, �e vzorka
$(x_1,\dots,x_n,\bf o)$ je rozbit� pomocou $P_n$. Dok�te, �e $\VCdim(G_n) = n$.
\end{enumerate}
\end{document}