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.

See also partial equivalence relation.
References in periodicals archive ?
This interpretive option was evaluated by the subsequent administration of the MTS-2a and MTS-2b tests of derived relations, which contained all baseline, symmetry, transitivity, and equivalence relations from the classes.
Assume that T is a nonempty closed subset of R and E is an equivalence relation on T.
If M1 = M2 = M, then both equivalence [equivalent to] and k-equivalence [equivalent to]k are equivalence relations that they obey the reflexive, symmetric and transitive laws.
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.
The authors attributed this difference to previously established behavioral relations interfering with the emergence of equivalence relations in the laboratory.
In the spirit of Saunders and Green's (1999) analysis of the effects of the kinds of discrimination involved in tests of stimulus equivalence formation in different training structures, it was felt that a gradual transition between the compound test and the matching-to-sample (MTS) test (described below) would maximize the chances of demonstrating stimulus equivalence relations in these two mutually confirmatory ways.
For example, Grey and Barnes (1996, Experiment 1) first trained and tested for the formation of three, three-member equivalence relations.
In addition to simulating the human data, the model was used to track a hypothetical timeline of developmental experience in order to map how increasing exposure to equivalence relations aids derived performance.
Sidman and Tailby (1982) argued that relations that possess the properties of reflexivity, symmetry, and transitivity constitute equivalence relations.
Equivalence relations are presented as examples of associative concepts.
Others state that the formation of equivalence relations requires no such mediation and that the formation of these classes plays a role in the development of language (Sidman, 1994), (see Fields & Nevin, 1993; Hayes, 1989; Horne & Lowe, 1996, 1997; and Sidman 1994 for reviews of this literature from differing viewpoints).
Although the basic phenomena of equivalence relations and derived relational phenomena had been described clearly in Skinner (1957), Sidman's research marked the beginning of a new and powerful empirical front in the analysis of such phenomena.