how many DFA can be constructed over the alphabet, such that no of states |Q|…

To determine the number of DFAs with |Q| = 2 and |∑| = 2 that accept the empty language (Φ), we need to ensure no string is accepted. This means all reachable states must be non-final, and there should be no path from the start state to any final state.

Step 1: There are 2 states (Q = {q0, q1}) and 2 input symbols (∑ = {a, b}). The number of possible transition functions is 2^(2×2) = 16, since each state has two transitions (one for a and one for b), and each transition can go to either of the 2 states.

Step 2: For each transition function, we analyze which states are reachable from the start state (q0). A state is reachable if there exists a string that leads to it from q0.

Step 3: For the language to be empty, no reachable state can be a final state. Therefore, we must choose final states only from unreachable states (if any). If all states are reachable, no final state can be selected.

Step 4: We classify transition functions based on reachability. For each function, we count valid final state assignments (subset of unreachable states) that result in an empty language. The total number across all functions is 20.

Final Answer: 20 DFAs can be constructed that accept the empty language.