\begin{frame}{Operations on Words}
  \begin{block}{Power}
    The power \alert{$v^k$} consists of $k$ concatenations of $v$'s:
    \begin{talign}
      v^0 &= \lambda  &
      v^{k+1} &= v^k v
    \end{talign}
  \end{block}
  
  \begin{exampleblock}{}
    Let $w = aba$. Then\vspace{-1ex}
    \begin{align*}
      w^0 &= \lambda &
      w^1 &= aba &
      w^2 &= abaaba &
      w^3 &= abaabaaba
    \end{align*}
  \end{exampleblock}
  \pause\smallskip
  
  \begin{block}{Reverse}
    The reverse of $a_1 \cdots a_n$ is 
    \begin{talign}
      \alert{(a_1 \cdots a_n)^R}=a_n \cdots a_1
    \end{talign}
    The reverse can be inductively defined
    \begin{talign}
      \lambda^R &= \lambda &
      (va)^R &= a(v^R)
    \end{talign}
  \end{block}
  
  \begin{exampleblock}{}
    The reverse of $abcb$ is $bcba$.
  \end{exampleblock}
\end{frame}

\subsection{Languages as Sets}

\themex{Languages}