High Quality Content by WIKIPEDIA articles! Generally, a trichotomy is a splitting into three disjoint parts. In mathematics, the law (or axiom) of trichotomy is most commonly the statement that for any (real) numbers x and y, exactly one of the following relations holds: x < y, x = y, x > y. Until the end of the 19th century the law of trichotomy was tacitly assumed true without having been thoroughly examined. A proof was sought by Logicians and the law was indeed proved to be true. If applied to cardinal numbers, the law of trichotomy is equivalent to the axiom of choice. 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.

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