Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.^[1] More generally, trichotomy is the property of an order relation $<$ on a set X that for any x and y, exactly one of the following holds: $x<y, x=y$, or $x>y$.

In mathematical notation, this is

$\forall x \in X \, \forall y \in X \, ( ( x < y \, \land \, \lnot (y < x) \, \land \, \lnot( x = y )\, ) \lor \, ( \lnot(x < y) \, \land \, y < x \, \land \, \lnot( x = y) \, ) \lor \, ( \lnot(x < y) \, \land \, \lnot( y < x) \, \land \, x = y \, \, ) ) \,.$

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

$\forall x \in X \, \forall y \in X \, ( (x < y) \, \lor \, (y < x) \, \lor \, (x = y) ) \,.$

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.

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).^[2]

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 element set {a,b,c} 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.

A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it is trivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

See also

• Begriffsschrift contains an early formulation of the law of trichotomy
• Dichotomy
• Law of noncontradiction
• Law of excluded middle

References

1. http://mathworld.wolfram.com/TrichotomyLaw.html
2. Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.