\begin{frame}{Exercises (1)} \begin{exampleblock}{} Show that $L = \{\, a^nb^n \mid n\geq 0 \,\}$ has the pumping property. \pause\medskip Let $m =2$. \pause Every word $w = a^n b^n$ with $|w| \ge m$ can be split \begin{talign} a^n b^n &= uvxyz & u &= a^{n-1} & v &= a & x &= \lambda & y &= b & z &= b^{n-1} \end{talign} \pause We have $|vxy| \le m$, $|vy| \ge 1$\pause and \begin{talign} uv^ixy^iz = a^{n-1+i} b^{n-1+i} \in L \end{talign} for every $i \ge 0$. \pause Thus the language has the pumping property. \end{exampleblock} \end{frame}