Law of trichotomy
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[1]
More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as <, this is stated in formal logic as:
- [math]\displaystyle{ \forall x \in X \, \forall y \in X \, ( [ x \lt y \, \land \, \lnot(y \lt x) \, \land \, \lnot(x = y) ] \, \lor \, [ \lnot(x \lt y) \, \land \, y \lt x \, \land \, \lnot(x = y) ] \, \lor \, [ \lnot(x \lt y) \, \land \, \lnot(y \lt x) \, \land \, x = y ] ) \,. }[/math]
Properties
- A relation is trichotomous if, and only if, it is asymmetric and connected.
- If a trichotomous relation is also transitive, then it is a strict total order; this is a special case of a strict weak order.[2][3]
Examples
- On the set X = {a,b,c}, the relation R = { (a,b), (a,c), (b,c) } is transitive and trichotomous, and hence a strict total order.
- On the same set, the cyclic relation R = { (a,b), (b,c), (c,a) } is trichotomous, but not transitive; it is even antitransitive.
Trichotomy on numbers
A law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"; some authors even fix y to be zero,[1] relying on the real number's additive linearly ordered group structure. The latter is a group equipped with a trichotomous order.
In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers.[clarification needed] The law does not hold in general in intuitionistic logic.[citation needed]
In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).[4]
See also
- Begriffsschrift contains an early formulation of the law of trichotomy
- Dichotomy
- Law of noncontradiction
- Law of excluded middle
- Three-way comparison
References
- ↑ 1.0 1.1 Trichotomy Law at MathWorld
- ↑ Jerrold E. Marsden & Michael J. Hoffman (1993) Elementary Classical Analysis, page 27, W. H. Freeman and Company ISBN:0-7167-2105-8
- ↑ H.S. Bear (1997) An Introduction to Mathematical Analysis, page 11, Academic Press ISBN:0-12-083940-7
- ↑ Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.
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