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:

a < b
a = b
a > b[2]

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

x R y
x = y
y R x

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

Properties of Trichotomous Relations
Symmetric property x R x is always false. A trichotomous relationship is not symmetric. For example, the statement 3 < 3 is always false.
Reflexive property If x R y then not y R x A trichotomous relationship is not reflexive. For example, 3 is less than 4, but 4 is not less than or equal to 3.
Transitive property If x R y and y R z then x R z A trichotomous relationship is typically transitive. For example, 3 < 4 and 4 < 5, then 3 < 5.
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  1. McAdams, David E.. All Math Words Dictionary, trichotomy property of real numbers. 2nd Classroom edition 20150108-4799968. pg 182. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. 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. Last Accessed 12/16/2018. Buy the book
  3. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 12-14. International Textbook Company. January 1963. Last Accessed 12/16/2018. Buy the book
  4. 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. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/16/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/18/2008: Initial version. (McAdams, David E.)

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