Problem 3. (30 points) a) (15 points) Prove using laws of propositional logic (not truth tables), the following: "if you study computer science, you'll be smart and if you study computer science, you'll be happy" is equivalent to "if you study computer science, you'll be smart and happy". Let p denote "you study computer science," q denote "you'll be smart" and "r" denote "you'll be happy". r). That is, b) (15 points) Prove, now using truth tables, that if (pq) and (qr), then (p prove that ((p q) ^ (g r)) (pr) is a tautology. This result (often used as a law of logic) is known as the "hypothetical syllogism