Order of Real Numbers is Dual of Order of their Negatives
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Theorem
- $\forall x, y \in \R: x > y \iff \paren {-x}< \paren {-y}$
Proof 1
Let $x > y$.
| \(\ds x\) | \(>\) | \(\ds y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + \paren {-x}\) | \(>\) | \(\ds y + \paren {-x}\) | Real Number Axiom $\R \text O1$: Usual Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(>\) | \(\ds y + \paren {-x}\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + \paren {-y}\) | \(>\) | \(\ds y + \paren {-x} + \paren {-y}\) | Real Number Axiom $\R \text O1$: Usual Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + \paren {-y}\) | \(>\) | \(\ds \paren {y + \paren {-y} } + \paren {-x}\) | Real Number Axiom $\R \text A1$: Associativity of Addition and Real Number Axiom $\R \text A2$: Commutativity of Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + \paren {-y}\) | \(>\) | \(\ds 0 + \paren {-x}\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-y}\) | \(>\) | \(\ds \paren {-x}\) | Real Number Axiom $\R \text A3$: Identity for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-x}\) | \(<\) | \(\ds \paren {-y}\) | Definition of Dual Ordering |
$\Box$
Let $\paren {-x} < \paren {-y}$.
| \(\ds \paren {-x}\) | \(<\) | \(\ds \paren {-y}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-x} + x\) | \(<\) | \(\ds \paren {-y} + x\) | Real Number Axiom $\R \text O1$: Usual Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(<\) | \(\ds \paren {-y} + x\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + y\) | \(<\) | \(\ds \paren {-y} + x + y\) | Real Number Axiom $\R \text O1$: Usual Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + y\) | \(<\) | \(\ds \paren {\paren {-y} + y} + x\) | Real Number Axiom $\R \text A1$: Associativity of Addition and Real Number Axiom $\R \text A2$: Commutativity of Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0 + y\) | \(<\) | \(\ds 0 + x\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(<\) | \(\ds x\) | Real Number Axiom $\R \text A3$: Identity for Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(>\) | \(\ds y\) | Definition of Dual Ordering |
$\blacksquare$
Proof 2
| \(\ds x\) | \(<\) | \(\ds y\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds y - x\) | \(>\) | \(\ds 0\) | Inequality iff Difference is Positive | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(>\) | \(\ds 0\) | Real Number Axiom $\R \text A2$: Commutativity of Addition | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds -x + -\paren {-y}\) | \(>\) | \(\ds 0\) | Negative of Negative Real Number | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds -x - \paren {-y}\) | \(>\) | \(\ds 0\) | Definition of Real Subtraction | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds -y\) | \(<\) | \(\ds -x\) | Inequality iff Difference is Positive |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(d)}$