Trichotomy

From Exampleproblems

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. Such a relation is called trichotomous. 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.