In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality.
(⧣ in unicode) to distinguish from the negation of equality (the denial inequality), which is weaker.
This last property is often called co-transitivity or comparison.
The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity.
is a tight apartness relation if it additionally satisfies: In classical mathematics, it also follows that every apartness relation is the complement of an equivalence relation, and the only tight apartness relation on a given set is the complement of equality.
are apart if there exists a rational number
The complex numbers, real vector spaces, and indeed any metric space then naturally inherit the apartness relation of the real numbers, even though they do not come equipped with any natural ordering.
Thus to say two real numbers are apart is a stronger statement, constructively, than to say that they are not equal, and while equality of real numbers is definable in terms of their apartness, the apartness of real numbers cannot be defined in terms of their equality.
For this reason, in constructive topology especially, the apartness relation over a set is often taken as primitive, and equality is a defined relation.
A set endowed with an apartness relation is known as a constructive setoid.
if the strong extensionality property holds This ought to be compared with the extensionality property of functions, i.e. that functions preserve equality.
Indeed, for the denial inequality defined in common set theory, the former represents the contrapositive of the latter.