To be admitted to the exam in December (15/12/), you must have at least 4+4 set out of the 12 assignments approved. At least 4 of the approved exercises sets must be from the first 6 sets, and at least 4 must be from the last 6 exercise sets. You are asked to pay attention to the quality of presentation, in particular, the cor- rectness of mathematical notation and formalism. Exercise 1. Let Pi and P2 be two posets on the sets X and Y, respectively. Consider the Cartesian product A = X x Y. The lexicographical order P on A is defined as follows: (x, y) P(x', y') is and only if x Pix' or (x = x') A (yPzy'). Show that P defines a poset on A. Exercise 2. Problems 9.9 and 9.11 on page 110 in Antonsen's textbook (Springer edition). Note: it will be helpful to read the corresponding section in Antonsen's textbook. Exercise 3. Problems 15.3, 15.5 on page 177 and problem 15.15 on page 178 in Antonsen's textbook (Springer edition). Note: it will be helpful to read the corresponding section in Antonsen's textbook. Exercise 4. Problem 12.3 on page 131 in the Lewis-Zax's textbook. 12.3. Consider the formula 3xayaz(P(x, y) A P(z, y) ~ P(x, z) A -P(Z, x)). Under each of these interpretations, is this formula true? In each case, R is the relation corresponding to P. (a) U= N, R= ((x, y) : x