8. In the theory of rational behavior, which has applications in economics, ethics and psychology, the notion of an individual preferring one object or state of affairs to another is of importance. We may say that an individual weakly prefers x to y if he does not strictly prefer y to x. We use the notion of weak preference for formal convenience, for if we use strict preferences, we also need a notion of indifference. The point of this question is to ask you to show that on the basis of two simple postulated properties of weak preference and the appropriate definition of strict preference and indifference in terms of weak preference, we may logically infer all the expected properties of strict preference and indifference. Let us use Q(x, y) to denote that individuals weakly prefer x over y, P(x, y) to denote that individuals strictly prefer x over y, and I(x, y) to denote that individuals are indifferent about their weak preference of x versus y. Our two premises on the predicate Q(x, y) just say that Q is transitive and that of any objects in the domain of objects under consideration, one is weakly preferred to the other. In Predicate Logic:

xyz((Q(x, y) Q(y, z)) Q(x, z))

xy(Q(x, y) Q(y, x))

As additional premises we introduce the two obvious definition for I and P.

xy(I(x, y) Q(x, y) Q(y, x))

xy(P(x, y) Q(y, x))

Derive the following conclusions from these four premises:

(a) xI(x, x).

(b) xy(I(x, y) I(y, x)).

(c) xyz((I(x, y) I(y, z)) I(x, z)).

(d) xy(P(x, y) P(y, x)).

(e) xy(I(x, y) ( P(x, y) P(y, x)))).

(f) xyz((I(x, y) P(y, z)) P(x, z)).