3. (a) Outline a sorting algorithm that is in (n). Give the algorithm and an explanation of why it is in (n). [15 marks] (b) Translate the following arguments into propositional judgements: i. If astrology is a true science, then the economy is improving. The economy is improving. Therefore, astrology is a true science. ii. If it is raining, then Bob stays at home. If it is raining then Alice stays at home. Therefore, if Alice and Bob are at home, then it is raining. iii. If it is raining, then Bob stays at home. If it is raining then Alice stays at home. Therefore, if it is raining, then Alice and Bob are at home. [15 marks] (c) Use the natural deduction rules to derive the following judgements: i. ((P and Q) implies R) if and only if (P implies (Q implies R)) ii. P implies Q, Q implies R+ P implies R [20 marks]