# Trichotomy Law

## 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.