a) Let R+ represent the collection of positive real numbers. Prove that R+ satisfies the following two properties.
i) For each x ∈ R, one and only one of the following holds:
x ∈ R+, -x ∈ R+, or x = 0.
ii) Given x, y ∈ 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) which satisfies properties i) and ii). Define x < y by y - x e R+. Prove that Postulate 2 holds with < in place of <.