Difference between revisions of "Equivalence relation"
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Revision as of 13:51, 6 March 2015
An equivalence relation is a special kind of relation
Required properties
Given a relation [ilmath]R[/ilmath] in [ilmath]A[/ilmath] we require the following properties to define a relation (these are restated for convenience from the relation page)
Reflexive
A relation [ilmath]R[/ilmath] if for all [ilmath]a\in A[/ilmath] we have [ilmath]aRa[/ilmath]
Symmetric
A relation [ilmath]R[/ilmath] is symmetric if for all [ilmath]a,b\in A[/ilmath] we have [ilmath]aRb\implies bRa[/ilmath]
Transitive
A relation [ilmath]R[/ilmath] is transitive if for all [ilmath]a,b,c\in A[/ilmath] we have [ilmath]aRb\text{ and }bRc\implies aRc[/ilmath]
Definition
A relation [ilmath]R[/ilmath] is an equivalence relation if it is:
- reflexive
- symmetric
- transitive
