(a) Show that the hypotheses "If you send me a WhatsApp message, then I will finish writing the Matlab program," "If you do not send me a WhatsApp message, then I will go to the cafe early" and "If I go to the cafe early, then I will come back feeling fresh" lead to the conclusion "If I do not finish writing the Matlab program, then I will come back feeling fresh." (8 marks) (b) Suppose you wish to use the Principle of Mathematical Induction to prove that n (n+1) for any positive integer n. 2 1+2+...+n= i) ii) iii) iv) v) Write P (1). Write P (6). Write P (k) for any positive integer k. Write P (k+1) for any positive integer k. Use the Principle of Mathematical Induction to prove that P(n) is true for all positive integer n. (c) Suppose that function f is defined recursively by f(0) = 3 f(n + 1) = 2 f(n) +3 for n = 0,1,2,... Find f(1), f(2), f(3) and f(4). (8 marks) (4 marks)