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.
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.
[clarification needed] The law does not hold in general in intuitionistic logic.