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.[1]
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. 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 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, with y = 0, where 0 is the zero of the integral domain or field.
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.
References
- ^ p148 Simon Singh Fermat's Last Theorem
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