\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}