11.

(a) Let R+ represent the collection of positive real numbers. Prove that R+

satisfies the following two properties.

(i) For each x E R, one and only one of the following hold:

xER+, -xER+, or x=O.

(ii) Given x, y E R+, both x + y and x . y belong to R+.

(b) Suppose that R contains a subset R+ (not necessarily the set of positive numbers) that satisfies properties (i) and (ii). Define x -< y by y - x E R + .

Prove that Postulate 2 holds with -< in place of <.