Question 1 (45 marks)a) For each of the following statements, write the converse, inverse and contrapositivestatement. Use p = hypothesis and q = conclusion. Also, provide the mathematicalnotations...

Question 1 (45 marks)a) For each of the following statements, write the converse, inverse and contrapositivestatement. Use p = hypothesis and q = conclusion. Also, provide the mathematicalnotations for the respective statements.(i) If the weather is nice, then I will wash the truck (5 marks)(ii) If � >5, then, �2>25 (5marks)(iii)~� →(~�∆�) (5marks)b) Write each of the following statements in the symbolic form and determine whichpairs (if any) are logically equivalent. Ensure that you explain the steps taken andthe reason for your decision.(i) Ifthegoodsareunsatisfactory,thenyourmoneywillberefunded.(ii) Goodssatisfactoryormoneyrefunded.(iii) Ifyourmoneyisnotrefunded,thenthegoodsaresatisfactory.(iv) Ifthegoodsaresatisfactory,thenyourmoneywillnotberefunded.(15 marks)c) Show that (p ∨ q) ∧ (r ∨ ¬ q) → (p ∨ r) is a tautology. Ensure that you explain thesteps taken and the reason for your decision. (15 marks)Question1AMarkingScheme• Explanationofeachstep– 5marks• Workings– 10marksQuestion1B MarkingScheme• Explanationofeachstep– 5marks• Workings – 10 marks2Question 2 (30 marks)Use mathematical induction to prove the following formulae for every positive integer na)11.2.3 +12.3.4 +13.4.5 + ⋯ +1�(� + 1)(� + 2) = �(� + 3)4(� + 1)(� + 2)Ensure that you explain the steps taken. (15 marks)b)1! + 3! + 5! + ⋯ + (2� − 1)! = �"(2�" − 1)Ensure that you explain the steps taken. (15 marks)Question2A MarkingScheme• Explanationofeachstep– 5marks• Workings – 10 marksQuestion2B MarkingScheme• Explanationofeachstep– 5marks• Workings – 10 marks