Trichotomy Law for Real Numbers
Theorem
The real numbers obey the trichotomy law.
That is, $\forall a, b \in \R$, exactly one of the following holds:
| \((1)\) | $:$ | $a$ is greater than $b$: | \(\ds a > b \) | ||||||
| \((2)\) | $:$ | $a$ is equal to $b$: | \(\ds a = b \) | ||||||
| \((3)\) | $:$ | $a$ is less than $b$: | \(\ds a < b \) |
Proof 1
This follows directly Real Numbers form Totally Ordered Field.
$\blacksquare$
Proof 2
$\le$ is a total ordering on $\R$.
The trichotomy follows directly from Trichotomy Law.
$\blacksquare$
Also known as
The trichotomy law can also be seen referred to as the trichotomy principle.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-2}$: Inequalities
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order properties (of real numbers): $(1)$ Trichotomy law
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trichotomy law
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers): $(1)$ Trichotomy law
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trichotomy law