Generally, a trichotomy is a splitting into three disjoint parts. A trichotomy is a splitting into three parts, and, apart from its normal literal meaning, can refer to: trichotomy (mathematics), in the mathematical field of order theory trichotomy (philosophy), for the idea that man has a threefold nature In taxonomy, a trichotomy is speciation of three groups from a common...

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: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

x < y,

x = y,

x > y.

If applied to cardinal numbers, the law of trichotomy is equivalent to the axiom of choice. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

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 orderhttp://en.wikipedia.org/wiki/Total_order#Strict_total_order; this is a special case of a strict weak order. For example, in the case of three elements 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 trichonomous relation. In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totally ordered ones are in black. ...

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, with y = 0, where 0 is the zero of the integral domain or field. In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

In set theory, trichotomy is most commonly defined as a property that a binary relation < has when all its members <x,y> satisfy exactly one of the relations listed above. Strict inequality is an example of a trichotomous relation in this sense. Trichotomous relations in this sense are irreflexive and antisymmetric. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... This article is about inequalities in mathematics. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ...