# Equivalence Relation

Also found in: Dictionary, Thesaurus, Medical, Wikipedia.
Related to Equivalence Relation: Equivalence class

## equivalence relation

[i′kwiv·ə·ləns ri′lā·shən]
(mathematics)
A relation which is reflexive, symmetric, and transitive. Also known as equals functions.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Equivalence Relation

a concept in logic and mathematics expressing the presence in different objects of the same characteristics or properties. The objects are indistinguishable—identical, equal, or equivalent—with respect to such shared characteristics. Any of them can serve equally well as a “representative” of the equivalence class to which all objects between which the equivalence relation holds belong. Equivalence relations are reflexive, symmetric, and transitive. Under certain conditions and within certain limits, they possess the property of substitution—that is, objects in an equivalence class can, with certain limitations, perform the same functions and their names, or words designating them, can be substituted for each other in different propositions.

## Equivalence Relation

an equality-type relation, that is, a binary relation that is reflexive, symmetric, and transitive. For example, if two geometric figures are congruent or similar or if two sets of objects are isomorphic or equipotent, the figures or sets are equal or identical in some regard. Thus, isomorphic sets are indistinguishable in structure if by “structure” is meant the aggregate of the properties with respect to which the sets are isomorphic.

## equivalence relation

(mathematics)
A relation R on a set including elements a, b, c, which is reflexive (a R a), symmetric (a R b => b R a) and transitive (a R b R c => a R c). An equivalence relation defines an equivalence class.

References in periodicals archive ?
Therefore, any equivalence relation on pitch or pitch-class sets other than transposition is necessarily ad hoc, at least in the context of the interval system.
In Section 2) we shall show that, in the Mal'cev context, algebraic exponentiation along split epimorphisms allows us to extend the existence of centralizers from subobjects to equivalence relations; accordingly, when the category C is moreover exact, we get a Schreier-Mac Lane extension theorem, according to [11].
The following theorem due to Hivert and Nzeutchap [13] shows that an equivalence relation on [A.sup.*] satisfying some properties can be used to define Hopf subalgebras of FQSym:
In particular, each function shows the likelihood of selecting the 1-node CA equivalence relation comparison plotted against the number of nodes characterizing the companion transitive relation.
Let U/R = {{[p.sub.1], [p.sub.2]}, {[p.sub.3]}} be an equivalence relation on U and A = {<[p.sub.1], (0.7, 0.6, 0.5) >, < [p.sub.2], (0.3, 0.4, 0.5) >, < [p.sub.3], (0.1, 0.5, 0.1) >} be a neutrosophic set on U then [N.bar](A) = {< [p.sub.1], (0.3, 0.4, 0.5) >, < [p.sub.2], (0.3, 0.4, 0.5) >, < [p.sub.3], (0.1, 0.5, 0.1) >}, [bar.N](A) = {<[p.sub.1], (0.7, 0.6, 0.5) >, < [p.sub.2], (0.7, 0.6, 0.5) >, < [p.sub.3], (0.1, 0.5, 0.1) >}, B(A) = {< [p.sub.1], (0.5, 0.6, 0.5) >, < [p.sub.2], (0.5, 0.6, 0.5) >, < [p.sub.3], (0.1, 0.5, 0.1) >}.
Assume that R is an equivalence relation, K = (U, R) is a knowledge base, and U/R = {[X.sub.1], [X.sub.2], ..., [X.sub.n]} is the equivalence class.
Hong and Do [24] improved this result and proposed a more refined equivalence relation. This equivalence relation can be used to partition the set of fuzzy numbers into equivalence class having the desired group properties for the addition operation.
It is also clear that in Perm, all preorders are equivalence relations: each of its components lies in some variety n-Perm, where it will be a congruence.
Because R is an equivalence relation on U, the classifications induced by R can be denoted as [[[x.sub.1]].sub.R], [[[x.sub.2]].sub.R], ..., [[[x.sub.n]].sub.R].
In Section 3, we discuss a canonical way to put an equivalence relation on P when it is a lattice and give three simple conditions which together imply that [chi](P, t) has nonnegative integral roots.
When M = M(C), the relation [R.sub.M(C)] is an equivalence relation on U.

Site: Follow: Share:
Open / Close