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:
![{\displaystyle \forall x\in X\,\forall y\in X\,([x<y\,\land \,\lnot (y<x)\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,y<x\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,\lnot (y<x)\,\land \,x=y])\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/419d5452ff0b5decba77dbc02d05ee38c55f2d86)
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
- ^ a b 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.