Trichotomy (mathematics)
In mathematics, the Law of Trichotomy states that every real number is either positive, negative, or zero.[1] More generally, trichotomy is the property of an order relation < on a set X that for any x and y, exactly one of the following holds: 
, 
, or 
.
In mathematical notation, this is
In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers.
In ZF set theory, the law of trichotomy holds between cardinal numbers if and only if the axiom of choice holds.
More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds. If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example, in the case of three element set {a,b,c} the relation R given by aRb, aRc, bRc is a strict total order, while the relation R given by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.
In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as more foundational than the law of total order.
Trichotomous relations cannot be reflexive, since xRx must be false. If transitive, they are trivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.