(a) Without using truth tables, show that ~(q^r) ~p is equivalent to (pq) ^ (~r ~p). [3 marks] (b) Given the hypotheses (qvd) ~m, ~k^~p, rv (z^m), rk, show that ~(d v k) is a valid conclusion. [5 marks] (c) Suppose that on an island there are three types of people, knights, knaves, and normal (also known as spies). Knights always tell the truth, knaves always lie, and normals sometimes lie and sometimes tell the truth. Detectives questioned three inhabitants of the island- Amy, Brenda, and Claire-as part of the investigation of a crime. The detectives knew that one of the three committed the crime, but not which one. They also knew that the criminal was a knight, and that the other two were not. Additionally, the detectives recorded these statements: Amy: "I am innocent." Brenda: "What Amy says is true." Claire: "Brenda is not a normal." After analyzing their information, the detectives positively identified the guilty party. Who was it? Explain how you arrived at your answer. [5 marks]