Problem 1. (25 points) Convert each of the following statements to an implication in propositional logic, and prove that the statement is true by proving that the implication is a tautology. (a) If you are sane, you are not dreaming if you can read this question. Therefore, if you are dreaming, you are insane if you can read this question. (b) If both Romeo and Juliet learn of each other's plan, then neither of them will die. If Romeo learns of Juliet's plan, but Juliet does not learn of Romeo's, then only Juliet will die, and vice versa. Therefore, if they both died, neither learned of the other's plan (c) If you are CS major then you study computer science and if you are a student of BU then you are currently in Boston. Therefore, if you are CS major then you are currently in Boston or if you are a student at BU then you study computer science