Ordering on Natural Numbers is Compatible with Multiplication/Corollary
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Corollary to Ordering on Natural Numbers is Compatible with Multiplication
Let $a, b, c, d \in \N$ where $\N$ is the set of natural numbers.
Then:
- $a > b, c > d \implies a c > b d$
Proof
| \(\ds a\) | \(>\) | \(\ds b\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds a c\) | \(>\) | \(\ds b c\) | Ordering on Natural Numbers is Compatible with Multiplication |
| \(\ds c\) | \(>\) | \(\ds d\) | ||||||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds b c\) | \(>\) | \(\ds b d\) | Ordering on Natural Numbers is Compatible with Multiplication |
Finally:
| \(\ds a c\) | \(>\) | \(\ds b c\) | from $(1)$ | |||||||||||
| \(\ds b c\) | \(>\) | \(\ds b d\) | from $(2)$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a c\) | \(>\) | \(\ds b d\) | Ordering on Natural Numbers is Trichotomy |
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 4$: The natural numbers: Exercise $1$