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:

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

With this definition, the law of trichotomy states that < is trichotomous relation on the set of real numbers. In other words, if x and y are real numbers, then exactly one of the following must be true: x<y, x=y, y<x.