2.225.62.162: /* Properties */

2013-10-27T09:35:44Z

Properties

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			In [[constructivism (mathematics)\|constructive mathematics]], an '''apartness relation''' is a constructive form of inequality, and is often taken to be more basic than [[equality (mathematics)\|equality]]. It is often written as # 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:
			# <math>\neg\;(x \# x)</math>
			# <math>x \# y \;\to\; y \# x</math>
			# <math>x \# y \;\to\; (x \# z \;\vee\; y \# z)</math>
			The [[negation]] of an apartness relation is an [[equivalence relation]], as the above three conditions become [[Reflexive relation\|reflexivity]], [[Symmetric relation\|symmetry]], and [[transitive relation\|transitivity]]. If this equivalence relation is in fact equality, then the apartness relation is called ''tight''. That is, # is a tight apartness relation if it additionally satisfies:
			:4. <math>\neg\;(x \# y) \;\to\; x = y</math>
			In [[classical logic\|classical]] mathematics, it also follows that every apartness relation is the negation of an equivalence relation, and the only tight apartness relation on a given set is the negation 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]] (one can construct) a [[rational number]] between them. In other words, real numbers ''x'' and ''y'' are apart if there exists a rational number ''z'' such that ''x'' < ''z'' < ''y'' or ''y'' < ''z'' < ''x''. The natural apartness relation of the real numbers is then the disjunction of its natural [[pseudo-order]]. 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.

			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 (mathematics)\|set]] is often taken as primitive, and equality is a defined relation.

			A set endowed with an apartness relation is known as a [[setoid\|constructive setoid]]. A function <math>f: A \rarr B</math> where ''A'' and ''B'' are constructive setoids is called a ''morphism'' for #<sub>''A''</sub> and #<sub>''B''</sub> if <math>\forall x, \, y: A.\, f(x) \; \#_B \; f(y) \Rarr x \; \#_A \; y</math>.

			{{DEFAULTSORT:Apartness Relation}}
			[[Category:Constructivism (mathematics)]]

en>Rgdboer: /* See also */ lk Matrix similarity

2012-08-07T23:59:59Z

Matrix equivalence - Revision history

2.225.62.162: /* Properties */

en>Rgdboer: /* See also */ lk Matrix similarity