Theory of computation

Using the Pumping Lemma, give formal proofs that the following languages are not regular/ regular:

This question further tests your understanding of the subtleties of the Pumping Lemma: (a) [5 marks] Let Bi = { 0kirot l k 21 andz e *). Give a formal proof that Bi is not a regular language. (b) [5 marks] Let B2 = {0krot I k > 1 and z e *). Show that B2 is a regular language. (c) We just proved in part (b) above that B2 is regular. Consider now the following argument, which claims to prove that B2 is, in fact, not regular. Assume B2 is regular, and let p be the pumping length. Consider string s0P10P E B2, and decompose it as s = zy2 with | = e, y 0p, 2-10p. Then, pumping s down by setting i = 0 yields string s' = zyiz = zyoz = 10p B2. Hence, by the Pumping Lemma, we have a contradiction. We conclude that B2 is not regular. i. [3 marks] What is wrong with this proof