Trichotomy Law (Ordering)

Theorem

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.

Proof

 $\ds \forall a, b \in S: \,$ $\ds$  $\ds a \preceq b \lor b \preceq a$ Definition of Total Ordering $\ds \leadstoandfrom \ \$ $\ds \forall a, b \in S: \,$ $\ds$  $\ds a \preceq b \lor a \succeq b$ Definition of Dual Ordering $\ds \leadstoandfrom \ \$ $\ds \forall a, b \in S: \,$ $\ds$  $\ds \paren {a = b \lor a \prec b} \lor \paren {a = b \lor a \succ b}$ Strictly Precedes is Strict Ordering $\ds \leadstoandfrom \ \$ $\ds \forall a, b \in S: \,$ $\ds$  $\ds a \prec b \lor a = b \lor a \succ b$ Rules of Commutation, Association and Idempotence

$\blacksquare$

Also known as

The Trichotomy Law can also be seen referred to as the trichotomy principle.