Surf Wiki
Save to docs
general/order-theory

From Surf Wiki (app.surf) — the open knowledge base

Law of trichotomy

Law (all real numbers are positive, negative, or 0)

Law (all real numbers are positive, negative, or 0)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.

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 :\forall x \in X , \forall y \in X , ( x [ \lnot(x [ \lnot(x ) ,. With this definition, the law of trichotomy states that In other words, if x and y are real numbers, then exactly one of the following must be true: x

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 total order; this is a special case of a strict weak order.

Examples

  • 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

In classical logic, this axiom of trichotomy holds for ordinary comparisons 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 for the cardinal numbers of well-orderable sets, but not necessarily for all cardinal numbers. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).

References

References

  1. [http://mathworld.wolfram.com/TrichotomyLaw.html Trichotomy Law] at [[MathWorld]]
  2. [[Jerrold E. Marsden]] & Michael J. Hoffman (1993) ''Elementary Classical Analysis'', page 27, [[W. H. Freeman and Company]] {{ISBN. 0-7167-2105-8
  3. H.S. Bear (1997) ''An Introduction to Mathematical Analysis'', page 11, [[Academic Press]] {{ISBN. 0-12-083940-7
  4. Bernays, Paul. (1991). "Axiomatic Set Theory". Dover Publications.
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

order-theory properties-of-binary-relations 3-(number)
Want to explore this topic further?

Ask Mako anything about Law of trichotomy — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report