请输入您要查询的字词:

 

单词 EquivalenceRelation
释义

equivalence relation


An equivalence relationMathworldPlanetmath on a set S is a relationMathworldPlanetmathPlanetmath that is:

ReflexiveMathworldPlanetmathPlanetmath.

aa for all aS.

Symmetric.

Whenever ab, then ba.

TransitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

If ab and bc then ac.

If a and b are related this way we say that they are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath under .If aS, then the set of all elements of S that are equivalent to a is called the equivalence classMathworldPlanetmath of a. The set of all equivalence classes under is written S/.

An equivalence relation on a set induces a partition on it. Conversely, any partition induces an equivalence relation. Equivalence relations are important, because often the set S can be ’transformed’ into another set (quotient spaceMathworldPlanetmath) by considering each equivalence class as a single unit.

Two examples of equivalence relations:

1. Consider the set of integers and take a positive integer m. Then m induces an equivalence relation by ab when m divides b-a (that is, a and b leave the same remainder when divided by m).

2. Take a group (G,) and a subgroupMathworldPlanetmathPlanetmath H. Define ab whenever ab-1H. That defines an equivalence relation. Here equivalence classes are called cosets.

Titleequivalence relation
Canonical nameEquivalenceRelation
Date of creation2013-03-22 11:48:27
Last modified on2013-03-22 11:48:27
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id15
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 06-00
Classificationmsc 03D20
Related topicQuotientGroup
Related topicEquivalenceClass
Related topicEquivalent
Related topicEquivalenceRelation
Related topicPartition
Related topicMathbbZ_n
Definesequivalent
Definesequivalence class
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 22:11:29