Apartness relation
In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as (⧣ in unicode) to distinguish from the negation of equality (the denial inequality) which is weaker. Description An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements are apart, then any other element is apart from at least one of them (this last property is often called co-transitivity or comparison). That is, a binary relation is an apartness relation if it satisfies: The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight. That is, is a if it additionally satisfies: 4. 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. So in that domain, the concept is not useful. In constructive mathematics, however, this is not the case. The prototypical apartness relation is that of the real numbers: