4. Regular expressions

4.1.1.

$ A=\{ 0^{2k} \mid k \geq 1 \}$

$ B = \{0, \epsilon \}$
$ AB = ? $

4.1.2. $ A = \{ 0^n 1^n \mid n \geq 1 \}$
$ B = \{ 1^n \mid n \geq 1 \} $
$ AB = ? $
$ BA = ? $

4.1.3. $ A = \emptyset $
$ B = \{ 1^n \mid n \geq 1 \} $
$ AB = ? $
$ A^* = ? $
$ B^* = ? $

4.3.1. Write a regular expression for the language of arithmetic expressions containing +, * and numbers. Hint: you can abbreviate $ 0 \cup 1 \cup \ldots \cup 9 $ by $ [0-9] $

4.3.2. Write a regular expression for $ L = \{ \omega \text{ in } \text{{0,1}} ^* \text{ | every sequence of consecutive zeros appears before ANY sequence of consecutive ones} \} $

4.3.3. Write a DFA for $ L(( 10 \cup 0) ^* ( 1 \cup \epsilon )) $

4.3.4. Write a regular expression which generates the accepted language of A:

4.3.5. Simplify the regular expression you found.

4.3.6. Describe as precisely as possible the language generated by $ (1 \cup 1(01^*0)1)^*$