[Solved]: Given 2 regular languages and their DFA’s, how to construct the DFA of the union?

Let’s denote the sets of states in $A_1$ and $A_2$ by $Q_1$ and $Q_2$, respectively. Furthermore, let’s denote by $F_1$ and $F_2$ the accepting states of these machines, $q_0’$ and $q_0”$ the start states of these machines, and $delta_1$ and $delta_2$ the transition functions of these machines. We can use the Cartesian-product-machine construction to define an automaton $A_3$ which accepts $L(A_1) cup L(A_2)$ as follows. First, let $Q_3 = Q_1 times Q_2$ be the set of states. Now, define $F_3 = {(q_1, q_2) in Q_3 mid q_1 in F_1 vee q_2 in F_2}$. Let $q_0”’$ be the start state and define $delta_3((q_1, q_2), sigma) = (delta_1(q_1, sigma), delta_2(q_2, sigma))$. Consider the functioning of such a machine. By virtue of its being in some state $(q_1, q_2) in Q_3$, we know by construction that $A_1$ would have be in state $q_1$ after seeing the same input, and we know $A_2$ would be in stare $q_2$ after seeing the same input. $A_3$ will accept if either $q_1$ was accepting in $A_1$ or $q_2$ was accepting in $A_2$, because of how we defined $F_3$. Intuitively, this works by demonstrating that we can model two DFAs executing in lockstep using the DFA formalism itself; two DFAs executing in lockstep is equivalent to the DFA constructed above. We have chosen to accept a string if either of the machines accepts, but as you can see, we could define a DFA for any of the standard binary set operations: union, intersection, difference, symmetric difference, etc.