Law of trichotomy

Inmathematics,thelaw of trichotomystates that everyreal numberis either positive, negative, or zero.

More generally, abinary relationRon asetXistrichotomousif for allxandyinX,exactly one ofxRy,yRxandx = yholds. WritingRas <, this is stated in formal logic as:

$\forall x\in X\,\forall y\in X\,([x<y\,\land \,\lnot (y<x)\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,y<x\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,\lnot (y<x)\,\land \,x=y])\,.$

Properties

• A relation is trichotomous if, and only if, it isasymmetricandconnected.
• If a trichotomous relation is also transitive, then it is astrict total order;this is a special case of astrict weak order.

Examples

• On the setX= {a,b,c}, the relationR= { (a,b), (a,c), (b,c) } is transitive and trichotomous, and hence a stricttotal order.
• On the same set, the cyclic relationR= { (a,b), (b,c), (c,a) } is trichotomous, but not transitive; it is evenantitransitive.

Trichotomy on numbers

A law of trichotomy on some setXof numbers usually expresses that some tacitly given ordering relation onXis a trichotomous one. An example is the law "For arbitrary real numbersxandy,exactly one ofx<y,y<x,orx=yapplies "; some authors even fixyto be zero,relying on the real number's additivelinearly ordered groupstructure. The latter is agroupequipped with a trichotomous order.

In classical logic, thisaxiom of trichotomyholds for ordinary comparison between real numbers and therefore also for comparisons betweenintegersand betweenrational numbers.The law does not hold in general inintuitionistic logic.

InZermelo–Fraenkel set theoryandBernays set theory,the law of trichotomy holds between thecardinal numbersof well-orderable sets even without theaxiom of choice.If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they areall well-orderablein that case).

See also

• Begriffsschriftcontains an early formulation of the law of trichotomy
• Dichotomy
• Law of noncontradiction
• Law of excluded middle
• Three-way comparison