theory of computing question

={1}, L = {w: :wis the unary encoding of 10, n 1}. For example, 1¹⁰ L; 1¹⁰⁰ L; 11 L.

Part 1. Use the RL pumping theorem to show that L RLs. Start by settingwto 1, where i is the first power of ten k, or 100 if k < 100. Then show that for all possible values ofy, pumping inyonce would generatew' L.

Part 2. Start the proof the same way as in part 1, and explain how the proof would fail if you chose to pump out once instead of in once.