In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
More generally, a binary relation R on some set X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds. Stated in formal logic for the case where the relation R is the order relation <:
Trichotomy on numbersEdit
A 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, 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.
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).
- Trichotomy Law at MathWorld
- Jerrold E. Marsden & Michael J. Hoffman (1993) Elementary Classical Analysis, page 27, W. H. Freeman and Company ISBN 0-7167-2105-8
- H.S. Bear (1997) An Introduction to Mathematical Analysis, page 11, Academic Press ISBN 0-12-083940-7
- Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.