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 QUESTIONS

Answer ALL Three Questions in This Section

START EACH NEW QUESTION ON A FRESH SHEET OF PAPER

Question A1 (3 marks total)

Let A. B be two logical statements. Which of the following three arguments are equivalent?

(i) B → (-A)

(ii) A → (-B)

(iii) (-A) v B

In your answer, provide the truth table corresponding to the arguments (i), (ii) and (iii) above.

Question A2 (3 marks total)

Can an incomplete preference relation be has a greatest element? Provide a proof of your argument.

Question A3 (3 marks total)

A decision maker has the following preferences on the set of all non-negative integer quantities of commodity 1 and commodity 2: α ≥ 夕 if and only if α 1 + α2  ≥ 夕1 + 夕2 .  If the decision maker has $5.5 income and the price of commodity 1 is $2 and commodity 2 is $1.  Find the set of greatest elements of the budget set of this agent: B = {α e Z+  × Z+ I2α1 + α2  ≤ 5u5}.