# Definition:Trichotomy

Jump to navigation
Jump to search

## Definition

Let $S$ be a set.

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

- $a \mathrel \RR b$
- $a = b$
- $b \mathrel \RR a$

## 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 if and only if $\prec$ is a
**trichotomy**.

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