\begin{frame}{Applications} \begin{exampleblock}{} $L \setminus \{\, \lambda \,\}$ is context-free for every context-free language $L$. \end{exampleblock} \pause\medskip \begin{exampleblock}{} $\{\,a^n b^n \mid n \geq 1000\,\}$ is context-free. \end{exampleblock} \pause\medskip \begin{exampleblock}{} Show that the language \begin{talign} L = \{\, w\in\{a,b,c\}^*\mid n_a(w) = n_b(w) = n_c(w) \,\} \end{talign} is \emph{not} context-free. \pause\medskip For a contradiction, assume $L$ was context-free. \pause\medskip The language $L(a^*b^*c^*)$ is regular, thus \begin{talign} L \cap L(a^*b^*c^*) = \{\, a^n b^n c^n \mid n\geq 0 \,\} \end{talign} would be context-free. \pause However, we know that it is not. \pause\medskip Contradiction. Thus $L$ is not context-free. \end{exampleblock} \end{frame}