\begin{frame}{Basic Properties of Context-Free Languages}
  \begin{alertblock}{}
    The intersection \alert{$L_1 \cap L_2$} is \alert{not} always context-free.\\
    (for context free languages $L_1$ and $L_2$)
  \end{alertblock}
  \pause\smallskip

  \begin{exampleblock}{}
    The languages $L_1$ and $L_2$ are context-free:
    \begin{talign}
      L_1 &= \{\, a^n b^n c^m \mid n \geq 0 \wedge m \geq 0 \,\} \\
      L_2 &= \{\, a^n b^m c^m \mid n \geq 0 \wedge m \geq 0 \,\}
    \end{talign}
    \pause
    However $L_1 \cap L_2 = \{\, a^n b^n c^n \mid n \geq 0 \,\}$ is \emph{not} context-free.
  \end{exampleblock}
  \pause\medskip
  
  \begin{alertblock}{}
    Also \alert{$\overline{L_1}$} and \alert{$L_1\backslash L_2$} are not always context-free.\\
    (for context free languages $L_1$ and $L_2$)
  \end{alertblock}  
  \pause\smallskip
  
  \begin{exampleblock}{}
    Namely, we have:
    \begin{talign}
      L_1 \cap L_2 &= \overline{\overline{L_1} \cup \overline{L_2}} &
      \overline{L_1} &= \Sigma^* \setminus L_1
    \end{talign}
  \end{exampleblock}
\end{frame}