Trichotomy Law (Ordering)

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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.


\(\ds \forall a, b \in S: \, \) \(\ds \) \(\) \(\ds a \preceq b \lor b \preceq a\) 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


Also known as

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