Equivalence Relation


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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.

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 ?
3 Parallel residues defining an equivalence relation
Definition 4 Let P be a poset and let ~ be an equivalence relation on P.
with respect to the triple (t, E, [alpha]) where T [subset equal to] R is a nonempty closed subset, E is the universal equivalence relation and [alpha] = sup t coincide with the corresponding ones with respect to the classical time scale, defined by Stefan Hilger [13,14,24-26].
From the perspective of RFT, the transformation of discriminative functions through equivalence relations is deemed to be functionally similar to the behaviour of a young child who responds appropriately to a symbol or word without a history of explicit training/and or reinforcement.
This was exactly the situation that these pre-service mathematics teachers experienced in the context of mathematical relations and equivalence relations.
Clearly, the delta relation is an equivalence relation in [\[Z.
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.
1 Let [equivalent to] be an equivalence relation defined on [A.
Most (if not all) do, on the other hand, define equivalence relation as a relation that is symmetric, reflexive, and transitive.
We are going to show that m is normal to the equivalence relation T and that U(T) [equivalent] R.
Equivalence relations are presented as examples of associative concepts.
Some theorize that language is a mediating variable in a process in which an equivalence relation is the outcome (e.