Using induction on

$$

$,$

$$prove that

$($

$$

$$

$)$

$$

$=$

$($

$$

$$

$)$

$$

$$for any string

$$

$$and all

$$

$>=$

$0$

$.$

Hints: feel free to use the following Theorem in your proof

Let

$$

$,$

vin

$$

$*$

$*$

$,$

$$then

$($

$$

$$

$)$

$$

$=$

$$

$$

$$

$$

$.$

For the following exercises, give a regular expression that represents that described set.

$2$

$.$

$$The set of strings over

$\{$

$$

$,$

$$

$,$

$$

$\}$

$$in which all the a

$\text{'}$

s precede the

$$

$\text{'}$

$$

$,$

$$which in turn precede the c

$\text{'}$

s

$.$

$$It is possible that there are no

$$

$\text{'}$

$$

$,$

$$

$\text{'}$

s

$,$

$$or c

$\text{'}$

s

$.$

$3$

$.$

$$The same as Exercise

$2$

$$without the null string.

$4$

$.$

$$The set of strings over

$\{$

$$

$,$

$$

$\}$

$$that contain the substring aa and the substring

$$

$$

$.$