DFSA

In some degree, the most challenging question in assignment 3 may be the last one.
I once got stuck on combing the two machines representing binary string with odd length and multiple of five respectively. I wasn't sure what were the exact steps using cartesian product. So i considered all the combinations of the cases of the third machine i want to construct.
That is exactly the result of cartesian product. Excluding the initial state, there are 5*2 = 10 states
for x = n mod 5 with odd/even length. These are the state invariants. Then We can see directly from some binary numbers and derive the transitions. Then the desired machine is fulfilled.
The proof of a DFA is straightforward. For me it is kind of a process of verifying each state go to the correct place via a path or an edge. Graphically, each state is a node with inputs and outputs.
Thus the transitions explain the correctness of traversals in a machine.