(Solved): Please answer, I will upvote! Question 4: In section 1.8.2, the text notes that the following expre ...

Please answer, I will upvote!

Question 4: In section 1.8.2, the text notes that the following expression [(P1 V P2 V V Pn) ?q] ? [(P1 ?g) ^ (p2 ?g) ^^ (Pn ?g)] is a tautology. This fact is used to justify the method of proof by cases. It is actually an infinite number of tautologies, one for each possible value of n. In this question we look more closely at how these tautologies are related to each other, in preparation for studying mathematical induction later in the course. (a) Consider the situation when n = 2, use a truth table to verify the tautology [(P1 V P2)?g] ? [(P? ? 9) ^ (P2 ?q)], and thus conclude that [(p1 V p2)?q] = [(P1 ?q)^(P2 ?q)]. Your truth table should have 8 rows. (b) We would like to verify the tautology when n = 3, i.e. the statement [(P1 V P2 V P3) ?q] ? [(P? ?g) ^ (P2 ?g) ^ (p3 ?g)], (†) but would prefer not to build a truth table with 16 rows. This exercise steps through this process. (i) Define r by r = (p? V p2). Show that the LHS of (†) is logically equivalent to [(r V p3) ?q]. (Hint: You probably want to use the associative property of V.) (ii) Apply (*) to the new expression, to find another equivalent expression, this time with (r ?q) as a subexpression. (iii) Replace r with (P? V p2) so that (*) can be applied again, this time to the subexpression. (iv) Use the associativity of A to conclude that what you have obtained is logically equivalent to the RHS of (†). (c) Now that you know (†) is a tautology, how much extra work would it be to verify the corresponding tautology for n = 4, n=5, or n = 782?