# Equivalence Relation

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Related to Equivalence relations: Equivalence classes

## equivalence relation

[i′kwiv·ə·ləns ri′lā·shən]
(mathematics)
A relation which is reflexive, symmetric, and transitive. Also known as equals functions.

## 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 ?
Class size effects on the establishing, maintenance, and recovery of equivalence relations.
Behavior researchers such as Sidman (1986) have proposed a coherent explanation of such expressions as "meanings, referent and rule-governed behavior" based on the establishment of relationships, mainly of equivalence relations, and which basically include what people say, and the reactions or responses to what is said.
K] is said to be a NP-soft equivalence relation if it is symmetric, transitive and reflexive.
Relating equivalence relations to equivalence relations: A relational framing model of complex human functioning.
One of the main results of [6] gives a sufficient condition: the Mal'tsev property, that is, 2-permutability RS = SR of internal equivalence relations R, S on the same object.
Assume that T is a nonempty closed subset of R and E is an equivalence relation on T.
In other words, in their view, it is because of naming that equivalence relations are observed.
The first published study designed to investigate the relationship between language and stimulus equivalence or derived equivalence relations was reported in the mid-eighties (Devany, et al.
of North Texas) presents a study of equivalence relations and classification problems that includes an introduction to basic concepts as well as a review of classical descriptions of set theory.
For example, Watt, Keenan, Barnes, and Cairns (1991) trained equivalence relations between Northern Irish Catholic names, nonsense syllables, and Northern Irish Protestant symbols to individuals with either a Protestant or Catholic upbringing from either Ireland or England.
Students from both Groups 1 and 2 were asked to write their reactions to the lesson on relations, their properties, and equivalence relations.
One of them results in a "new" definition of equivalence relations, and consequently a new representation for them, that seems to be overlooked by the experts.

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