Law of trichotomy

In 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:

${\displaystyle \forall x\in X\,\forall y\in X\,([x<y\,\land \,\lnot (y<x)\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,y<x\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,\lnot (y<x)\,\land \,x=y])\,.}$

Properties

• A relation is trichotomous if, and only if, it is asymmetric and connected.
• If a trichotomous relation is also transitive, then it is a
  strict weak order.^[2][3]

• On the set X = {a,b,c}, the relation R = { (a,b), (a,c), (b,c) } is transitive and trichotomous, and hence a strict total order.
• On the same set, the cyclic relation R = { (a,b), (b,c), (c,a) } is trichotomous, but not transitive; it is even
  antitransitive
  .

Trichotomy on numbers

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,^[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

• Begriffsschrift contains an early formulation of the law of trichotomy
• Dichotomy
• Law of noncontradiction
• Law of excluded middle
• Three-way comparison

References