For each of the following languages over the alphabet {0, 1}, give a regular expres- sion...

 

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For each of the following languages over the alphabet {0, 1}, give a regular expres- sion that describes that language, and briefly argue why your expression is correct. (a) All strings that begin with 1010101 and have length divisible by 5. (b) All strings that do not begin with 101 or 010, and end with 00110. (c) All strings x such that the number of "leading zeros" in x is divisible by 3, and x contains an odd number of ones. (For example, the string 00000101001 has 5 leading zeros and is not in the language; the string 101001 has 0 leading zeros and is in the language.) (d) All strings that have an even number of occurrences of 01 as a substring. (For example, 11110000 and 0000111101111 are in the language but 0000111100011100011 is not.) For each of the following languages over the alphabet {0, 1}, give a regular expres- sion that describes that language, and briefly argue why your expression is correct. (a) All strings that begin with 1010101 and have length divisible by 5. (b) All strings that do not begin with 101 or 010, and end with 00110. (c) All strings x such that the number of "leading zeros" in x is divisible by 3, and x contains an odd number of ones. (For example, the string 00000101001 has 5 leading zeros and is not in the language; the string 101001 has 0 leading zeros and is in the language.) (d) All strings that have an even number of occurrences of 01 as a substring. (For example, 11110000 and 0000111101111 are in the language but 0000111100011100011 is not.)