Let F be a field and suppose that P is a subset of F that satisfies the three properties in Exercise. Define x < y iff y − x ∈ P. Prove that “ < ” satisfies axioms O1, O2, and O3. Thus in defining an ordered field, either we can begin with the properties of “ < ” as in the text, or we can begin by identifying a certain subset as “positive.”
Exercise
Let P ={ x ∈ ℝ:x > 0}. Show that P satisfies the following:
(a) If x, y ∈ P, then x+y ∈ P.
(b) If x, y ∈ P, then x · y ∈ P.
(c) For each x ∈ ℝ, exactly one of the following three statements is true:
x ∈ P, x = 0, − x ∈ P.
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