Trichotomy Property of Real Numbers

Pronunciation: /trɪˈkɒt ə mi ˈprɒ pər ti ʌv ˈriəl ˈnʌm bərz/ Explain

The trichotomy property of real numbers states that, for any two real numbers a and b, exactly one of the following is true:

For any equivalence relation R on set A, the relation is trichotomous if for all x and y in A exactly one of

xRy, x=y, or yRx
holds.

A trichotomous relation is not symmetric, not reflexive, but is transitive.

Properties of Trichotomous Relations
PropertyEquationDescription
Symmetric propertyxRx is always falseA trichotomous relationship is not symmetric. For example, the statement 3<3 is always false.
Reflexive propertyif xRy then not yRxA trichotomous relationship is not reflexive. For example, 3 is less than 4, but 4 is not less than 3.
Transitive propertyif xRy and yRz then xRzA trichotomous relationship is typically transitive. For example, 3<4, 4<5, and 3<5.
Table 1

References

  1. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pg 3. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 12/19/2009. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n12/mode/1up. Buy the book
  2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 12-14. www.archive.org. International Textbook Company. January 1963. Last Accessed 1/12/2010. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book
  3. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 58-59. Thomson, Brooks/Cole. 2005. Buy the book

Cite this article as:

McAdams, David E. Trichotomy Property of Real Numbers. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/t/trichotomy.html.

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Revision History

12/19/2009: Added "References". (McAdams, David E.)
12/18/2008: Initial version. (McAdams, David E.)

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