suppose M1 and M2 are two DFA's withe the same input alphabet.We'll refer to the state set, start set, final state set, and transition function of M1 as S1, r1,F1, and b1 respectively, and similarily for M2,

we define the product DFA M1*M2 as follows, the state set is the direct product S1*S2, the set of ordered pair (s1,s2) with s1S1 and s2S2. the start state is the pair (r1,r2), and the final state set is F1*F2. the new transition function takes a state (s1,s2) and a letter a to {b1(s1,a),b2(s2,a)}

prove that the DFA decides the language L(M1) Union L(M2)