Question

3) Construct a regular expression defining each of the following languages over the alphabet {a, b}.

(a) L = {aab, ba, bb, baab};

(b) The language of all strings containing exactly two b's.

(c) The language of all strings containing at least one a and at least one b.

(d) The language of all strings that do not end with ba.

(e) The language of all strings that do not containing the substring bb.

(f) The language of all strings in which every b is followed immediately by aa.

Answer 1

a) The regular expression is: (aab + ba + bb + baab)

b) The regular expression is: (a*ba*ba*)

c) The regular expression is: (a + b)*a⁺b⁺(a + b)*

d) The regular expression is: (a + b) * (aa + ba + bb) + a + 1 + ε

NOTE: As per Chegg policy, I am allowed to answer only 4 questions (including sub-parts) on a single post. Kindly post the remaining questions separately and I will try to answer them. Sorry for the inconvenience caused.

Answer 2

(a) L = {aab, ba, bb, baab}

Regular expression: aab|ba|bb|baab

Explanation: The regular expression consists of four options separated by the pipe symbol "|". Each option represents one string from the language L. The first option "aab" matches the string "aab," the second option "ba" matches the string "ba," the third option "bb" matches the string "bb," and the fourth option "baab" matches the string "baab."

(b) The language of all strings containing exactly two b's.

Regular expression: (aba*)b(aba*)

Explanation: This regular expression matches all strings that contain exactly two b's. The pattern consists of three parts: (aba*), b, and (aba*). The first and last parts, (aba*), match any number of 'a's (including zero) before and after the 'b', allowing the 'b' to appear anywhere in the string. The middle part, 'b', ensures that the string contains exactly one 'b'.

(c) The language of all strings containing at least one a and at least one b.

Regular expression: (a|b)(a+b)(a|b)

Explanation: This regular expression matches all strings that contain at least one 'a' and at least one 'b'. The first part, (a|b), matches any combination of 'a' and 'b' (including an empty string). The second part, (a+b), ensures that the string contains at least one 'a' and at least one 'b'. The third part, (a|b), matches any remaining combination of 'a' and 'b' after the occurrence of 'a' and 'b'.

(d) The language of all strings that do not end with 'ba'.

Regular expression: (a|b)*(b|ε)

Explanation: This regular expression matches all strings that do not end with 'ba'. The first part, (a|b)*, matches any combination of 'a' and 'b' (including an empty string). The second part, (b|ε), matches either 'b' or an empty string (ε), allowing the string to end with either 'b' or nothing.

(e) The language of all strings that do not contain the substring 'bb'.

Regular expression: (a|ε)b?(a|ε)

Explanation: This regular expression matches all strings that do not contain the substring 'bb'. The first part, (a|ε), matches any combination of 'a' (including an empty string). The second part, b?, matches zero or one occurrence of 'b', allowing the string to contain at most one 'b'. The third part, (a|ε), matches any combination of 'a' (including an empty string) after the occurrence of 'b'.

(f) The language of all strings in which every 'b' is followed immediately by 'aa'.

Regular expression: (a|b|aa)*

Explanation: This regular expression matches all strings in which every 'b' is followed immediately by 'aa'. The expression (a|b|aa) matches any combination of 'a', 'b', or 'aa'. The asterisk (*) allows the expression to repeat zero or more times, meaning any combination of these elements can appear any number of times in the string.