Please provide the correct answers to all the 3 questions with explanation.

1. (10 pts.) In the following questions, you must provide proofs for your an swers. If you believe that some statement is false, give a counter example You may make use of commonly believed assumptions, like P NP, or NP co-NP Let L1 and L2 be arbitrary languages in NP. Which of the following statements are true? L1 U L2 is in NP . L1 n L2 is in NP L,\L2 is in NP 2. (10 pts.) Now assume that L1 and L2 are NP-complete languages. Which of the following statements are true? L1 U L2 is N P-complete Ln L2 is N P-complete . Li\L2 is NP-complete 3. (5 pts.) Let S, be the language of Boolean SAT, restricted to i-variables For each i EN, Si is a different language What do you think of the complexity of the Si? . Are the Si in NP? Are Si in P? . Are the S, NP-complete