Definition:Trichotomy

Definition

Let $S$ be a set.

A trichotomy on $S$ is a relation $\mathcal R$ on $S$ such that for every pair of elements $a, b \in S$, exactly one of the following three conditions applies:

• $a \mathcal R b$
• $a = b$
• $b \mathcal R a$

Example

A classic example of a trichotomy is the standard less than ordering on the set of real numbers.

Also see

• Trichotomy Law: an ordering $\prec$ is a strict total ordering iff $\prec$ is a trichotomy.