Inconstructive mathematics, anapartness relation is a constructive form of inequality, and is often taken to be more basic thanequality.

An apartness relation is often written as${\displaystyle \mathrm{\#}}$ (⧣ inunicode) to distinguish from the negation of equality (thedenial inequality), which is weaker. In the literature, the symbol${\displaystyle \ne }$ is found to be used for either of these.

A binary relation${\displaystyle \mathrm{\#}}$ is an apartness relation if it satisfies:^[1]

1. ${\displaystyle \mathrm{\neg }(x\mathrm{\#}x)}$
2. ${\displaystyle x\mathrm{\#}y\to y\mathrm{\#}x}$
3. ${\displaystyle x\mathrm{\#}y\to (x\mathrm{\#}z\vee y\mathrm{\#}z)}$

So an apartness relation is asymmetricirreflexivebinary 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 calledco-transitivity orcomparison.

Thecomplement of an apartness relation is anequivalence relation, as the above three conditions becomereflexivity,symmetry, andtransitivity. If this equivalence relation is in fact equality, then the apartness relation is calledtight. That is,${\displaystyle \mathrm{\#}}$ is atight apartness relation if it additionally satisfies:

4.${\displaystyle \mathrm{\neg }(x\mathrm{\#}y)\to x=y.}$

Inclassical 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: two real numbers are said to be apart ifthere exists (one can construct) arational number between them. In other words, real numbers${\displaystyle x}$ and${\displaystyle y}$ are apart if there exists a rational number${\displaystyle z}$ such that or The natural apartness relation of the real numbers is then the disjunction of its naturalpseudo-order. Thecomplex numbers, realvector spaces, and indeed anymetric space then naturally inherit the apartness relation of the real numbers, even though they do not come equipped with any natural ordering.

If there is no rational number between two real numbers, then the two real numbers are equal. Classically, then, if two real numbers are not equal, one would conclude that there exists a rational number between them. However it does not follow that one can actually construct such a number. 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, inconstructive topology especially, the apartness relation over aset is often taken as primitive, and equality is a defined relation.

A set endowed with an apartness relation is known as aconstructive setoid. A function${\displaystyle f:A\to B}$ between such setoids${\displaystyle A}$ and${\displaystyle B}$ may be called amorphism for${\displaystyle {\mathrm{\#}}_A}$ and${\displaystyle {\mathrm{\#}}_B}$ if the strong extensionality property holds

${\displaystyle \mathrm{\forall }(x,y:A).f(x){\mathrm{\#}}_{B}f(y)\to x{\mathrm{\#}}_{A}y.}$

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.

• Equivalence class – Mathematical concept

• Binary relations
• Constructivism (mathematics)