# Definition:Trichotomy

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## 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:

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

## 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**. - Results about
**trichotomies**can be found here.