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 Ll be the language L1 = {a'b' | i 20 and p is a prime}. Prove that the language L2 = b* U L, satisfies the conditions of the Pumping Lemma. I.e. show that there exists a q N such that for every word w L2 with |w| 2q we can write w = ryz such that cyl 0, and for every i > 0, ry'z 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 ly2q there are strings a, b, c such that y = abc, |6| > 0, and for all i > 0, xab'cz E L. (c) (10 points) Prove that the language L2 is not regular