Law of trichotomy

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x=''y holds. Writing R'' as <, this is stated in formal logic as:
With this definition, the law of trichotomy states that < is a trichotomous relation on the set of real numbers.
In other words, if x and y are real numbers, then exactly one of the following must be true: x<''y, x''=y, y<''x''.

Properties

A relation is trichotomous if, and only if, it is asymmetric and connected.
If a trichotomous relation is also transitive, then it is a strict total order; this is a special case of a strict weak order.

Examples

On the set X =, the relation R = is transitive and trichotomous, and hence a strict total order.
On the same set, the cyclic relation R = is trichotomous, but not transitive; it is even antitransitive.

Trichotomy on numbers

A law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"..