Trichotomy Property of Real Numbers

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

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

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

Properties of Trichotomous Relations
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


  1. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd edition, pg 3. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19).
  2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 12-14. International Textbook Company, January 1963. (Accessed: 2010-01-12).
  3. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 58-59. Thomson, Brooks/Cole, 2005.

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Trichotomy Property of Real Numbers. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC.


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

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