Several common rules of inference are given below. p+q p-q P qr (Hypothetical Syllogism) ..por pVq P P-9 79 P (Modus Ponens) (Modus Tollens) 8.9 (Disjunctive Syllogism) P ..pVq P 9 p^q (Addition) (Conjunction) (a) Use rules of inference to construct a valid argument that uses the premises p, pq, rq to reach the conclusion -r. In each step, state explicitly what rule of inference you are using. (b) You are given the following three premises: (i) There is a lifeguard shortage. (ii) If there is a lifeguard shortage, then the pools are closed. (iii) If the waterslide is running, then the pools are open. Do you have enough information to logically conclude whether or not it the waterslide is running? Explain how to reach a conclusion, or describe what extra information you require. Hint: assign appropriate propositional variables, and use the conclusion of part (a).