Trichotomy Law (Ordering)

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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

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

$\blacksquare$


Also known as

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


Sources