Prove that A={re {a,b)* | Each a in z is followed by at least one b) is regular.

Introduction:

To prove that the language A = {x ⋵ (a, b)* | Each a in x is followed by at least one b} is regular, we can construct a deterministic finite automaton (DFA) that accepts this language.

Let us build the DFA as follows:

Begin with the first state, q0.

Create a transition from the current state to the next state q1 for each input symbol a.

Create a transition from the current state to the next state q2 for each input symbol b.

Create a loop transition for input symbol b to return to the same state q1, and a transition for input symbol a to return to the initial state q0 for state q1.

Create a loop transition for input symbol b to return to the same state q2, and a transition for input symbol a to return to the beginning state q0 for state q2.

The last stage is q2, which supports the language A.