equivalence relation

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equivalence relation

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equivalence relation

Definition/Summary

A relation

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is an equivalence relation if it is reflexive, symmetric and transitive. In other words, if
1)

(the relation is reflexive)
2)

(the relation is symmetric)
3)

(the relation is transitive)

The equivalence class containing

is the set of all elements related to

An equivalence relation divides its set into distinct equivalence classes.

The set of all equivalence classes of a set is its quotient set

Equations

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Breakdown

Mathematics
> Foundations
>> Set Theory

See Also

Relation

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Extended explanation

Commentary

qspeechc @ 01:16 PM Oct30-09

I agree with matt grime that this can be written better. People who will want to know the definition of a relation will be students just starting out with abstract mathematics, and so a gentler, less formal, etc. definition is needed in my humble opinion.

tiny-tim @ 05:35 PM Mar10-09

looks ok to me

and being a subset of AxA is mentioned, on the first line

matt grime @ 05:11 PM Mar10-09

Boy, does this need rewriting.

Examples would be good.

Mentioning that = is an equivalence relation would seem to be a good idea.

Not using the definition of a 'relation' as being a subset of AxA, which it then confusingly doesn't explain before the list of axioms.