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Law of trichotomyIn mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[1] More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as <, this is stated in formal logic as: With this definition, the law of trichotomy states that < is trichotomous relation on the set of real numbers. In other words, if x and y are real numbers, then exactly one of the following must be true: x<y, x=y, y<x. Properties
Examples
Trichotomy on numbersA law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"; some authors even fix y to be zero,[1] relying on the real number's additive linearly ordered group structure. The latter is a group equipped with a trichotomous order. 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.[clarification needed] The law does not hold in general in intuitionistic logic.[citation needed] In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).[4] See also
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