Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero. More generally, trichotomy is the property of an order relation ${\displaystyle <}$ on a set X that for any x and y, exactly one of the following holds: ${\displaystyle x<y,x=y}$, or ${\displaystyle x>y}$.

In mathematical notation, this is

Assuming that the ordering is irreflexive and transitive, this can be simplified to

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. The law does not hold in general in intuitionistic logic.

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