Real Number is Greater than Zero iff its Negative is Less than Zero
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Theorem
- $\forall x \in \R: x > 0 \iff \paren {-x} < 0$
Proof
Let $x > 0$.
\(\ds x\) | \(>\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x + \paren {-x}\) | \(>\) | \(\ds 0 + \paren {-x}\) | Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(>\) | \(\ds 0 + \paren {-x}\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(>\) | \(\ds \paren {-x}\) | Real Number Axiom $\R \text A3$: Identity for Addition |
$\Box$
Let $\paren {-x} < 0$.
\(\ds \paren {-x}\) | \(<\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x + \paren {-x}\) | \(<\) | \(\ds x + 0\) | Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(<\) | \(\ds x + 0\) | Real Number Axiom $\R \text A4$: Inverses for Addition | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(<\) | \(\ds x\) | Real Number Axiom $\R \text A3$: Identity for Addition |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(c)}$