Need step in detail handwriting is accepted but make sure wo

Need step in detail, handwriting is accepted but make sure words clear to understand the solution.

Give regular expressions for the following languages: {w epsilon{a, b}* | w has an even number of b\'s} {w epsilon {a, b}* | w does not contain the substring ab} {w epsilon (a, b)* | w contains substring ab an even number of times} {w epsilon{a, b}* | w has an even number of a\'s and even number of b\'s} To help the grader out (and to increase chance of partial credit), if you have along regular expression, please identify small conceptual parts and explain what each part does.

Solution

1.a)

(a*ba*ba*)⁺

1.b)

b*a*

1.c)

((a⁺b⁺)(a⁺b⁺))⁺

1.d)

Regualr expression : (aa + bb)* ((ab+ba) (aa+bb)* (ab+ba) (aa+bb)*)*