If a equal to b then not a is equal to not b?

No because if a=4 and b=4 then a=b and not a can equal 6 and not b can equal 5 so ~a is not = to ~b

the Answer is Sometimes

Yes. I'm assuming this is talking asking about boolean logic (the question makes little sense otherwise). If a and b are equal, then the complement of a and the complement of b are equal.

A is therefore equal to C.

Yes. A = B = C