\begin{frame}{Exercises (2)}

  \begin{exampleblock}{}
    Show $L = \{\, w \in \{a,b\}^* \mid w=w^R \,\}$ for has the pumping property.
    \pause\medskip
    
    Let $m =3$. \pause Every word $w \in L$ with $|w| \ge m$ has the form
    \begin{talign}
      w = sctcs^R
    \end{talign}
    where $s \in \{a,b\}^*$, $c \in \{a,b\}$ and $t \in \{\, a,b,\lambda \,\}$.
    \pause
    Thus
    \begin{talign}
      w &= uvxyz & 
      u &= s &
      v &= c &
      x &= t &
      y &= c &
      z &= s^R
    \end{talign}
    \pause
    We have $|vxy| \le m$, $|vy| \ge 1$\pause and 
    \begin{talign}
      uv^ixy^iz = s c^{i} t c^{i} s^R \in L
    \end{talign}
    for every $i \ge 0$.
    \pause
    Thus the language has the pumping property.
  \end{exampleblock}
  \pause
  
  \begin{exampleblock}{}
    Show that $L$ also has the pumping property for $m = 2$.\\[1ex]
    \emph{Hint:} distinguish $w$ of even and odd length when splitting.
  \end{exampleblock}
\end{frame}