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
Property | Equation | Description |
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
- 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
- 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
- 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
- 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. http://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.)
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.)