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
Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
) to distinguish from the negation of equality (the ''denial inequality'')
which is weaker.
Description
An apartness relation is a
symmetric irreflexive
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal ...
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of an apartness relation is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
, as the above three conditions become
reflexivity,
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, 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: two real numbers are said to be apart if
there exists
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
(one can construct) a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
between them. In other words, real numbers
and
are apart if there exists a rational number
such that
or
The natural apartness relation of the real numbers is then the disjunction of its natural
pseudo-order. The
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, real
vector spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, and indeed any
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
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, in
constructive topology especially, the apartness relation over a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is often taken as primitive, and equality is a defined relation.
A set endowed with an apartness relation is known as a
constructive setoid. A function
where
and
are constructive setoids is called a ''morphism'' for
and
if
See also
*
References
{{DEFAULTSORT:Apartness Relation
Binary relations
Constructivism (mathematics)