2. (10 points) In class we have seen that context-free languages are not closed under intersection. In this problem, we will show that if L is a context-free language and K is a regular language then LNK is context-free. a) Let MI = (Q1, E, I, 81, 91, F1) be a PDA that recognizes L and M2 = (Q2, 2, 82, 92, F2) a DFA that recognizes K. Consider the construction of a PDA that recognizes LNK. N = (Q x Q2, E, I, 8, (q1, q2), Fix F2). with transition function ( { (a, 82(r, u)), B) | (a, B) E 81((a, u, v))} if ufe d((a,r), u, v) = { (a,r), B ) | (a, B) E 81((q, u, v))} if u = for qe Q1,re Q2, u E Ee, v ere. Apply this construction to the PDA and DFA below: 0,;0 1.0 1,0; b) Describe using set builder notation the language that is recognized by the PDA given in part a. (no justification necessary.) c) Describe using set builder notation the language that is recognized by the DFA given in part a. (No justification necessary.) d) Describe using set builder notation the language that is recognized by the PDA that is the result of the construction in part a. (You must describe it without using the intersection operation. No justification necessary.)