Generally, a trichotomy is a splitting into three disjoint parts.

In mathematics, the most concrete **law of trichotomy** that is usually seen is the statement that for any real numbers *x* and *y*, *exactly* one of the following relations holds: *x* < *y*, *x* = *y*, *x* > *y*.

More generally, a law of trichotomy is any statement that for some binary relation on some set *S*, which we may denote by using the "less than" symbol "<", and for any two members *x*, *y* ∈ *S*, exactly one of the relations above holds. For a transitive binary relation this is exactly equivalent to saying that the binary relation in question is a linear ordering of the set *S*.

In the special case of cardinal numbers, trichotomy is equivalent to the Axiom of Choice.

In taxonomy a trichotomy is speciation of three groups from a common ancestor, where it is unclear or unknown in what chronological order the three groups split.