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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:

  • $a \mathop {\mathcal R} b$
  • $a = b$
  • $b \mathop {\mathcal R} a$


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

Also see