\begin{frame}{Operations on Languages}
  \begin{alertblock}{}
    Attention: $L^2 = \{uv \mid u,v \in L \} \neq \{ uu \mid u \in L \}$\; !
  \end{alertblock}
  \pause\bigskip
  
  \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 $L^*=L^+\cup\{\lambda\}$.
  
  \begin{exampleblock}{}
    Let $L = \{\, a,bb \,\}$. Then
    \begin{talign}
      L^* = \{\, \lambda, a, bb, aa, abb, bba, bbbb, aaa, aabb, abba, abbbb \ldots\,\}
    \end{talign}
  \end{exampleblock}  

  \begin{goal}{}
    $L^*$ are all the words that you can build from `\textbf{building blocks}' $L$.
  \end{goal}
  \bigskip
\end{frame}