Operations on Alphabet, Languages, or Strings

• Length: $\vert S\vert$ is number of characters in $S$.
• Concatenation: $w = a_1a_2 \ldots a_n$, and $u= b_1b_2 \ldots b_m$ then $wu =a_1a_2 \ldots a_n b_1b_2 \ldots b_m$. $L_1L_2 = \{ xy \vert x \in L_1, y \in L_2 \}.$
• Reverse: $w = a_1a_2 \ldots a_n$ then $w^R = a_na_{n-1} \ldots a_1$.
• Power:
  
  $$\Sigma^n = \{ w \vert \vert w\vert=n \}.$$
  
  
  
  
  $$\Sigma^* = \bigcup_{n \ge 0} \Sigma^n.$$
  
  
  
  
  $$\Sigma^+ = \bigcup_{n \ge 1} \Sigma^n.$$
  
  
  The second is also called Kleene Star.
  
  $$L^n = \{ w=w_1w_2\ldots w_m \ \vert \ w_i \in L,\ 1 \le i \le n \ \}.$$
  
  
  
  
  $$L^* = \bigcup_{n \ge 0} L^n.$$
  
  
  
  
  $$L^+ = \bigcup_{n \ge 1} L^n.$$
  
  

• complement: $\overline{L} = \Sigma^* -L$.
• Suffix, prefix:

Therefore we can order the strings in a languages. Typically, we can use the lexicographic order.