\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}