Question 2 (12 marks) Part (a) involves propositional logic only, so you should not use any quantifiers. a) Alice, Bob, and Carol are trying to organise a lunch together for Tuesday. Let A denote the proposition \"Alice is available", and B and C similarly for Bob and Carol. For each of the five statements below, write a propositional expression in conjunctive normal form (CNF) that is equivalent to the statement. i. All three of Alice, Bob, and Carol are available. ii Atleast one of Alice, Bob, or Carol is available. iii, | At most one of Alice, Bob, or Carol is available. (Hint: There are three possible pairs of people: (Alice, Bob), (Alice, Carol), (Bob, Carol). For each pair, what can we definitely say about those two people's (non-)availabilities?) iv. Exactly one of Alice, Bob, and Carol is available. (Hint: How does this statement relate to the previous two statements?) v. Atleast two of Alice, Bob, and Carol are available. (Hint: Again consider the three pairs of people.) (5x 1=5 marks) Parts (b) and onward now involve predicate logic. There are different subsets of the natural numbers, including the square numbers (1, 4,9, 16,...), cube numbers (1, 8,27, 64,...) and triangular numbers (1,3, 6, 10,...). Define the following predicates: e Square(x) means that x is a square number. * Cube (a) means that x is a cube number. e Triangular (a) means that x is a triangular number. Greater (x,y) means thatx > y. For the following parts you will assume the domain to be the natural numbers; hence you can just write e.g. da and Vy instead of ax N and vy N. You must use the predicates defined above where relevant. b) Write a corresponding predicate sentence for: \"There exists a square triangular number.\" (1 mark) c) Write a corresponding predicate sentence for: \"Given any square number s and any larger cube number c (i.e. > s), there exists a triangular number in between them (ie.s