PLEASE USE PROOF OF INDUCTION. THIS IS THE SECOND TIME IM UPLOADING. TIA

Argue that each of the following languages is regular by showing how to construct it using the definition of regular languages: a. (2 marks) The language of all binary strings of even length. b. (2 marks) The language of strings over the alphabet {a,b,c} in which every a is followed by a c. So the string acbbbacccbacbccb is in the language because every a is followed by a c. The strings abcacccbc and cbbcaaccacbb are not in the language because of the highlighted occurrences of a that are not followed by a c. C. (2 marks) The language of all binary strings with an odd number of Os or a number of 1s that is divisible by 3. The strings 011001110110, 011001101001, and 0110011010010 belong to the language: 011001110110 contains an odd number of Os, 011001101001 contains a number of 1s divisible by 3, and 0110011010010 satisfies both conditions. The string 01101001100 is not in the language because it contains an even number of Os and the number of 1s is not divisible by 3. highlighted occurrences of a that are not followed by a c. Example: The language of all strings over the alphabet {a,b,c} that contain the substring aaa is regular: We start with our base cases: The languages {a}, {b}, and {c} are regular. This implies that the language {aaa} = {a}{a}{a} is regular, as is the language {a,b,c} = {a} U {b} U {c}. The latter implies that {a,b,c}* is regular. Therefore, the language {a,b,c}*{aaa}{a,b,c}* is regular. But this is exactly the desired language because any string in this language consists of an arbitrary number of letters, followed by the substring aaa, followed by another group of an arbitrary number of letters, that is, it contains the substring aaa