230 \end{tabular}

   230 \end{tabular}

   231 \end{center}

   231 \end{center}

   232 

   232 

   233 \noindent Note the difference in the last two lines: the empty

   233 \noindent Note the difference in the last two lines: the empty

   234 set behaves like $0$ for multiplication and the set $\{[]\}$

   234 set behaves like $0$ for multiplication and the set $\{[]\}$

   236 

   236 

   237 Following the language concatenation, we can define a

   237 Following the language concatenation, we can define a

   238 \defn{language power} operation as follows:

   238 \defn{language power} operation as follows:

   239 

   239 

   240 \begin{eqnarray*}

   240 \begin{eqnarray*}

   245 \noindent This definition is by recursion on natural numbers.

   245 \noindent This definition is by recursion on natural numbers.

   246 Note carefully that the zero-case is not defined as the empty

   246 Note carefully that the zero-case is not defined as the empty

   247 set, but the set containing the empty string. So no matter

   247 set, but the set containing the empty string. So no matter

   248 what the set $A$ is, $A^0$ will always be $\{[]\}$. (There is

   248 what the set $A$ is, $A^0$ will always be $\{[]\}$. (There is

   249 another hint about a connection between the $@$-operation and

   249 another hint about a connection between the $@$-operation and

   253 $A^*$ is the union of all powers of $A$, or short

   254 $A^*$ is the union of all powers of $A$, or short

   254 

   255 

   255 \begin{equation}\label{star}

   256 \begin{equation}\label{star}

   256 A^* \dn \bigcup_{0\le n}\; A^n

   257 A^* \dn \bigcup_{0\le n}\; A^n

   257 \end{equation}

   258 \end{equation}