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\begin{frame}{Operations on Languages (2)}
  \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}
  \pause
  
  \begin{alertgoal}{}
    Attention: $L^2 = \{uv \mid u,v \in L \} \neq \{ uu \mid u \in L \}$
  \end{alertgoal}
  \pause\medskip
  
  \begin{block}{Kleene star}
    \begin{malign}
      \alert{L^*} &\;\;=\;\; \bigcup_{i=0}^\infty \; L^i \;\;=\;\; L^0 \cup L^1 \cup L^2 \cup L^3 \cup \cdots\\
      \alert{L^+} &\;\;=\;\; \bigcup_{i=1}^\infty \; L^i \;\;=\;\; L^1 \cup L^2 \cup L^3 \cup \cdots
    \end{malign}
  \end{block}
  Thus $L^*=L^+\cup\{\lambda\}$.
\end{frame}