Regular Expression to Derive Strings with Even Occurrences of the Substring "ab"

The regular expression that derives all the strings of "as" and "bs" where each string contains even occurrences of the substring "ab" is:

a) (b abab)*

Let's break down the regular expression and explain each part:

(b abab)*

1. (b: Matches the character 'b' literally.

2. : Represents concatenation, indicating that the characters before and after it should appear in that order.

3. abab: Matches the characters 'abab' literally, representing the substring "ab" occurring twice.

4. ): Closes the parentheses.

5. *: Matches the preceding element zero or more times.

Therefore, the regular expression can be read as "zero or more occurrences of the concatenation of 'b' followed by the substring 'abab'."

Explanation:

The regular expression ensures that each string contains an even number of occurrences of the substring "ab" by utilizing the concept of repetition.

1. The expression starts with the character 'b', ensuring that every string begins with a 'b'.

2. The expression then specifies that the substring "abab" should occur zero or more times. This allows for any number of 'as' and 'bs' to be present in between the occurrences of "abab".

3. Since the substring "abab" occurs twice in the expression, any string derived from this regular expression will have an even number of occurrences of "ab".

Example:

Let's consider a few examples to see how the regular expression works:

- "b" satisfies the regular expression as it starts with 'b' and has zero occurrences of "abab".

- "bab" satisfies the regular expression as it starts with 'b' and has one occurrence of "abab".

- "babab" satisfies the regular expression as it starts with 'b' and has two occurrences of "abab".

- "bababab" satisfies the regular expression as it starts with 'b' and has three occurrences of "abab".

- "bbababab" satisfies the regular expression as it starts with 'b' and has four occurrences of "abab".

As all the examples illustrate, the regular expression ensures that each string contains an even number of occurrences of the substring "ab", making option a) the correct answer.