Inequalities

Definition

ℝ⁺ = {a | a ∈ ℝ and a is located to the R.H.S. of 0 in the number line}
ℝ^- = {a | a ∈ ℝ and a is located to the L.H.S. of 0 in the number line}
{0} = The single point 0

Then, ℝ = ℝ^- ∪ {0} ∪ ℝ⁺

Definition

Let a, b ∈ ℝ

1. a - b ∈ ℝ⁺, We say that a is greater than b. When this happens, we write: a > b
2. a - b ∈ ℝ⁺ ∪ {0}, We say that a is greater than or equal to b. When this happens, we write: a ≥ b
3. a - b ∈ ℝ^-, We say that a is less than b. When this happens, we write: a < b
4. a - b ∈ ℝ^- ∪ {0}, We say that a is less than or equal to b. When this happens, we write: a ≤ b

r ∈ ℝ⁺ ⇒ r - 0 ∈ ℝ⁺ ⇒ r > 0
r ∈ ℝ⁺ ∪ {0} ⇔ r ≥ 0
r ∈ ℝ^- ⇔ r < 0
r ∈ ℝ^- ∪ {0} ⇔ r ≤ 0

Convection

a ∈ ℝ⁺ & b ∈ ℝ⁺ ⇒ ab ∈ ℝ⁺
a ∈ ℝ⁺ & b ∈ ℝ^- ⇒ ab ∈ ℝ^-
a ∈ ℝ^- & b ∈ ℝ⁺ ⇒ ab ∈ ℝ^-
a ∈ ℝ^- & b ∈ ℝ^- ⇒ ab ∈ ℝ⁺

Theorem

Let a, b, c ∈ ℝ

1. a > b ⇒ ra > rb ∀r > 0 & ra < rb ∀r < 0
2. a ≥ b ⇒ ra ≥ rb ∀r > 0 & ra ≤ rb ∀r < 0
3. a < b ⇒ ra < rb ∀r > 0 & ra > rb ∀r < 0
4. a ≤ b ⇒ ra ≤ rb ∀r > 0 & ra ≥ rb ∀r < 0
5. a > b & b > c ⇒ a > c
6. a < b & b < c ⇒ a < c
7. a > b ⇒ a + d > b + d & a - d > b - d ∀d ∈ ℝ
8. a ≥ b ⇒ a + d ≥ b + d & a - d ≥ b - d ∀d ∈ ℝ
9. a < b ⇒ a + d < b + d & a - d < b - d ∀d ∈ ℝ
10. a ≤ b ⇒ a + d ≤ b + d & a - d ≤ b - d ∀d ∈ ℝ

Proof:

Let a, b ∈ ℝ
a > b ⇒ ra > rb ∀r > 0 & ra < rb ∀r < 0

Let a > b & r > 0

Then a - b ∈ ℝ⁺ & r ∈ ℝ⁺
r (a - b) ∈ ℝ⁺ [Convection]
ra - rb ∈ ℝ⁺ [Distributive Law]
ra > rb [by Definition]

Now let a > b & r < 0

Then a - b ∈ ℝ⁺ & r ∈ ℝ^-
r (a - b) ∈ ℝ^- [Convection]
ra - rb ∈ ℝ^- [Distributive Law]
ra < rb [by Definition]