Trichotomy Law
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Theorem
General Ordering
Let $\struct {S, \preceq}$ be an ordered set.
Then $\preceq$ is a total ordering if and only if:
- $\forall a, b \in S: \paren {a \prec b} \lor \paren {a = b} \lor \paren {a \succ b}$
That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element.
In other words, if and only if $\prec$ is a trichotomy.
Integral Domain
The property:
- $\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$
is known as the trichotomy law.
Real Numbers
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 $b$: | \(\ds a < b \) |
Natural Numbers
Let $\omega$ be the set of natural numbers defined as the von Neumann construction.
Let $m, n \in \omega$.
Then one of the following cases holds:
- $m \in n$
- $m = n$
- $n \in m$
Also known as
The Trichotomy Law can also be seen referred to as the trichotomy principle.