2 Strict preference relations In class we've built preferences up from a weak preference relation, which indicates whether one outcome is "as good as" another. In this problem we'll explore an alternative formulation, in which we describe whether one outcome is "better than" another. Fix an outcome set X. A binary relation R on X is asymmetric if R(, y) mplies R(y,z) = 0. A binary relation is negatively transitive if R(x,y) = 0 and R(y, z) = 0 implies R(,z) 0 I. Suppose X = {a,b,c). Exhibit a binary relation on X which is: (a) Asymmetric and negatively transitive (b) Asymmetric but not negatively transitive (c) Negatively transitive but not asymmetric. (d) Neither negatively transitive nor asymmetric. 2. Describe in words why asymmetry and negative transitivity are reasonable properties to demand of a binary relation indicating whether one outcome is strictly better than another. 3. Suppose that R is asymmetric and negatively transitive. Define a new binary relation R" by Ru(z,v) = 1 iff R(y, z) = 0. Show that Rio is complete and transitive. How would you describe in words the statement "R(a, y) 1