Argue that each of the following languages is regular by showing how to construct it using the definition of regular languages: 1) The language of all binary strings of even length. Please prove by direct proof.

Please proof the other two using induction or contradiction.

b. (2 marks) The language of strings over the alphabet {a,b,c} in which every b is immediately followed by two c s. The string acbccbccbccaacccbcccbcc is in this language because every b is followed by two c s (more than two is fine; there just have to be two c s immediately after the b ). The string accacccbcc is not in this language because the highlighted occurrence of b is followed by only one c. c. (2 marks) The language of all binary strings with an even number of Os or a number of 1s that is divisible by 4 . The strings 0110011101010,001100110111001 , and 1011001101100100 belong to this language: 0110011101010 contains an even number of Os, 001100110111001 contains a number of 1s divisible by 4, and 1011001101100100 satisfies both conditions. The string 001101001100 is not in this language because it contains an odd number of Os and the number of 1s is not divisible by 4