\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}