Trichotomy (mathematics)
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- For other uses, see trichotomy (disambiguation).
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.
If applied to cardinal numbers, the law of trichotomy is equivalent to the axiom of choice.
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.
