DO NOT COPY ANY OTHER ANSWERS OFF CHEGG OR I WILL DOWNVOTE! 1. (40 points) In this problem we investigate the limits of the Pumping Lemma as it was stated in class and look for an alternative that remedies one of these shortcomings. (a) (10 points) Let L, be the language L1 = {ab' | i 20 and p is a prime}. Prove that the language L2 = b*UL, satisfies the conditions of the Pumping Lemma. I.e. show that there exists a q E N such that for every word w E L2 with |w|2q we can write w = xyz such that|xy 0, and for every i 2 0, xyz L2. (b) (20 points) Prove the following generalization of the Pumping Lemma: Let L be a regular language. There exists a q E N such that for every w E L and every partition of w into w = xyz with \yl 2 q there are strings a, b, c such that y = abc, 160 > 0, and for all i > 0, cabcz L. (C) (10 points) Prove that the language L, is not regular