Order Property of ℝ

 Order Property of ℝ

1. Given \(x, y ∈ ℝ\), with \(x < y\) then  

a) \(∀ c ∈ ℝ, (x + c < y + c) \)    b) \(∀ c ∈ ℝ,c > 0(x . c < y . c) \)

2. Transitive: 

\(∀(x, y, z) ∈ ℝ × ℝ × ℝ\), with \(x < y\) and \(y < z\) then \(x < z\)

3. Law of Trichotomy: \(∀(x, y) ∈ ℝ × ℝ\) exactly one of the following holds 

(a) \(x < y\) (b) \(x = y\) (c) \(x > y\)