For the language over Σ = {a, b}, provide a regular expression and a DFA for the language. Your automata should have at most 10 states.The set of strings that contain exactly three as and an even number of bs. For example, abababb is in the language, while ababab is not.

Regular Expression: The regular expression for the language that contains exactly three 'a's and an even number of 'b's can be represented as:

(bababab)(b(bb))

Explanation:

• b* represents zero or more occurrences of 'b'.
• a represents exactly one occurrence of 'a'.
• (bababab) represents exactly three 'a's with any number of 'b's before, between, and after them.
• (b(bb)*) represents an even number of 'b's.

DFA (Deterministic Finite Automaton): The DFA can be represented with 8 states (q0 to q7).

• q0: Start state. Transition to q1 on 'a', remain at q0 on 'b'.
• q1: Transition to q2 on 'a', transition to q0 on 'b'.
• q2: Transition to q3 on 'a', transition to q1 on 'b'.
• q3: Transition to q4 on 'b', remain at q3 on 'a'.
• q4: Transition to q3 on 'b', remain at q4 on 'a'.
• q5, q6, q7: Dead states. Any input leads to these states.

The final state is q4, which represents exactly three 'a's and an even number of 'b's. Any string that leads to a state other than q4 is not accepted.