# 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
holds.

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

Properties of Trichotomous Relations
PropertyEquationDescription
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.
Table 1

### References

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. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 12/16/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n12/mode/1up. Buy the book
3. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 12-14. www.archive.org. International Textbook Company. January 1963. Last Accessed 12/16/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. 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

McAdams, David E. Trichotomy Property of Real Numbers. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/t/trichotomy.html.

### 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.)