1. (6 pts) Given that Vertex Cover is an NP Complete problem, prove that Set Cover is an NP Complete problem.

a. (1 pt) Explain the optimization versions of Set Cover and Vertex Cover.

b. (1 pt) Explain the decision versions of both problems. c. (3 pts) Create an appropriate transformation to show that Set Cover is at least as difficult as Vertex Cover (Warning: You should reduce the known NP-complete problem to the problem you are interested in)

Steps you should follow:

i. Generate an illustrative example to show your transformation from one instance to another.

ii. Generalize your example to cover any arbitrary instances.

iii. Prove your transformation meets all of Toveys 3 Criteria

d. (1 pt) Prove Set Cover is in NP 2.