Problem 1 Consider the universal set U = {a,b,c,aa, 1,2,5). Given the two sets A = {a,b, aa, 5), B = {5.c, b), calculate: i. AUB AB A-B iv. B-A v. vi. B vii. AUB viii. AB Problem 2 i. Consider the set S = {a,b). Write down explicitly the powerset of S, P, (or 2$ following your textbook's notation). ii. How many partitions of Sexist? Write down all possible partitions. iii. Write down the cartesian product set S x S. Problem 3 i Consider a vertex set V = {a,b,c,d,e). Draw your own directed graph for this set Vand write down your edge set E. 11. Define/show a walk, a path, and a simple path on your graph. ili. Does your graph have a cycle? iv. From your graph, isolate a subgraph that is a tree. Problem 4 Consider the alphabet ? = {a,b,c). Derive formally a grammar G that produces the language L(G) = {(ab)"c": n 20). Problem 5 What language is generated by the grammar G = ({S, A,B),(a, b),S,P) where the productions P are S AB; A - a; B BBb12