a) A is regular , A U B is regular , then B is not necessary regular
example :- A = (a+b)* B = a^{n}b^{n} n>=0 A U B is (a+b)* while B is not regular.
b ) we have two machine M 1 and M 2
draw a DFA using M1 and M2 where start state is, say, p0q0 (where p0 is start state in M1 and q0 is start state in M2)
∂(p0q0, 0) = ∂(p0, 0) U ∂(q0, 0)
if L(M1) ⊆ L (M2)
Then final state of M1 will come together with final state of M2, while Final state of M2 can come alone.
i.e All inputs of M1 is also in machine M2 , and there may be different inputs in M2.