8.1.1. Show that the pumping lemma holds for finite languages.
8.1.2.* Find a language which is not regular for which the pumping lemma holds.
Show that each of the languages from the list below is not regular.
8.2.1. $ L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $
8.2.2. $ L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $
8.2.3. $ L = \{(01)^n(10)^n \mid n > 0 \} $
8.2.4. $ L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $
8.2.5. $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $
8.2.6. $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a power of two} \: \} $
8.2.7. $ L = \{ \: ww^R \: | \: w\in \{0,1\}^* \} $
8.3.1. Using the pumping lemma, prove that $ L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language.