\begin{frame}{Operations on Languages}
  \begin{block}{Concatenation}
    The concatenation of languages $L_1$ and $L_2$ is defined as
    \begin{talign}
      \alert{L_1 L_2} &= \{\, xy \mid x \in L_1 \wedge y \in L_2 \,\}
    \end{talign}
  \end{block}
  
  \begin{exampleblock}{}
    Let $L_1 = \{\, a,\; bb \,\}$ and $L_2 = \{\, ab,\; ba \,\}$. Then
    \begin{talign}
      L_1 L_2 = \{\,aab,\; aba,\; bbab,\; bbba\,\}
    \end{talign}
  \end{exampleblock}
  \pause\medskip
  
  \begin{block}{Power}
    The $n$-th power of a language $L$ is defined by induction on $n$:
    \begin{talign}
      \alert{L^0} &= \{\, \lambda \,\} &
      \alert{L^{n+1}} &= L^n L \qquad (n \geq 0)
    \end{talign}
  \end{block}
  
  \begin{exampleblock}{}
    Let $L = \{\, a,\; bb \,\}$. Then
    \begin{talign}
      L^2 &= \{\, aa, abb, bba, bbbb\,\}\\
      L^3 &= \{\, aaa, aabb, abba, abbbb, bbaa, bbabb, bbbba, bbbbbb \,\}
    \end{talign}
  \end{exampleblock}
\end{frame}