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# Trichotomy

<https://en.wikipedia.org/wiki/Trichotomy_(mathematics>)

The law of trichotomy states that every real number is either positive, negative, or zero.

More generally, a binary relation `R` on a set `X` is **trichotomous** if `∀x∀y ∈ X`, exactly one of `xRy`, `yRx` and `x = y` holds.

Properties

* A relation is trichotomous iff it is *asymmetric* and *semi-connex*
* If a trichotomous relation is also *transitive*, then it is a *strict total order*; this is a special case of a *strict weak order*.

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*.

In classical logic, the axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers. The law does not hold in general in intuitionistic logic.

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).