There are a total of five (5) questions in this examination, three (3) in Section A and two (2) in Section B. Answer ALL questions in Sections A and B. Justify all answers. There is a total of 20 marks.  Each question carries the number of marks as indicated.  Partial credit may be awarded if a substantial part of the answer or working is provided.  You need not waste time explaining your notations as long as you are using the notations we have developed in class.

Section A: SHORT ANSWER QUESTIONS

Answer ALL Three Questions in This Section

Question A1 (3 marks total)

Let A, B, C be three logical statements.  Assume that A is True, B is False and C is True.  Find the truth value of the statement (A A (-B)) e (C ^ B), i.e., whether it is True or False.  In your answer, show your work by deriving the truth values of the statements (-B), A A (-B), and C ^ B . [Note that you do not have to provide full truth table.]

Question A2 (4 marks total)

Assume that Alice is the minister of health in her country. There is an infection virus affecting her country and Alice needs to choose one of the five vaccines X = {A, B, C, D, E} to fight with the virus. Assume that the country has enough resources to buy any of the vaccines and the supply of each vaccine is enough to satisfy the demand.  Alice has ordered vaccines A and B, and does not order other vaccines. Assume that Alice is benevolent and chooses the best option for the country. Alice mentions in a press conference that  “The vaccines I ordered are not inferior options among the available alternatives.” Which of the following are implied by this information?

(a) [2 marks] Alice’s preference relation on vaccines has to be complete and transitive.

(b) [2 marks] Vaccines C, or D, or E cannot be a greatest element of X  based on Alice’s

preference relation.

In your answer,  explain why the claim  in each  part above  is  implied  by or  not  implied  by the information provided.  That it, if it is implied, provide a proof of your argument, and if it is not implied, then pride an example supporting your argument.

Question A3 (3 marks total)

Let X = [0, 1].  Consider the following claim:  “For every complete binary relation on a non-empty set X, there exists a subset A of X that has a greatest element.” Is this statement true or false? If true provide a proof, if false provide a counterexample.