Answer:
Step-by-step explanation:
(a) L1:
To construct a regular expression for L1, we need to break it down into its components.
L1 consists of strings of the form uvuR, where u and v are any strings over the alphabet {a, b} and |u| = 2.
The regular expression for L1 can be written as:
(a|b)(a|b)(a|b)*(a|b)(a|b)\2\1
Step-by-step explanation:
(a|b)(a|b) matches the requirement of |u| = 2.
(a|b)*(a|b) matches the possible values of v.
\2 refers to the second occurrence of the captured group, which represents v.
\1 refers to the first occurrence of the captured group, which represents u.
(b) L2:
To construct a regular expression for L2, we need to ensure that there are no consecutive 'a's or consecutive 'b's in the string.
The regular expression for L2 can be written as:
((ab|ba)(a|b))
Step-by-step explanation:
(ab|ba) matches either 'ab' or 'ba', ensuring no consecutive 'a's or consecutive 'b's at the start.
(a|b)* allows for any sequence of 'a's or 'b's after the initial pattern.
((ab|ba)(a|b)) repeats this pattern zero or more times to match the entire string.
(c) L3:
To construct a regular expression for L3, we need to cover two conditions:
When n(w) is divisible by 3.
When w contains the substring 'bb'.
The regular expression for L3 can be written as:
(a|b)(bbb)(a|b)(bbb)((a|b)(a|b)(a|b))|(a|b)(bbb)*
Step-by-step explanation:
(a|b)(bbb)(a|b)(bbb) matches the condition of w containing the substring 'bb'.
((a|b)(a|b)(a|b))* matches the condition of n(w) being divisible by 3.
| denotes the OR operation, so we account for either condition.
Please note that there can be variations or alternative expressions for each language, but the provided regular expressions should cover the specified languages.