Difference between revisions of "Equivalence relation"
m (→See Also: Adding canonical map) 
m (Fixing mathematical notation typo) 

Line 17:  Line 17:  
! 2  ! 2  
 [[Relation#Types_of_relationSymmetric]]   [[Relation#Types_of_relationSymmetric]]  
−   {{M\forall x,y\in X[  +   {{M\forall x,y\in X[(x,y) \in \sim \implies (y,x) \in \sim]}}. Which we write {{M\forall x,y \in X[x\sim y \implies y\sim x]}}. 
    
! 3  ! 3 
Latest revision as of 15:18, 12 February 2019
This page is waiting for a final review, then this notice will be removed.
Contents
Definition
A relation, [ilmath]\sim[/ilmath], in [ilmath]X[/ilmath]^{[Note 1]} is an equivalence relation if it has the following properties^{[1]}:
Name  Definition  

1  Reflexive  [ilmath]\forall x\in X[(x,x) \in \sim][/ilmath]. Which we write [ilmath]\forall x\in X[x\sim x][/ilmath]. 
2  Symmetric  [ilmath]\forall x,y\in X[(x,y) \in \sim \implies (y,x) \in \sim][/ilmath]. Which we write [ilmath]\forall x,y \in X[x\sim y \implies y\sim x][/ilmath]. 
3  Transitive  [ilmath]\forall x,y,z\in X[((x,y) \in \sim \wedge (y,z) \in \sim) \implies (x,z) \in \sim][/ilmath]. Which we write [ilmath]\forall x,y,z \in X [(x\sim y \wedge y\sim z) \implies x\sim z][/ilmath]. 
Terminology
 An equivalence class is the name given to the set of all things which are equivalent under a given equivalence relation.
 Often denoted [ilmath][a][/ilmath] for all the things equivalent to [ilmath]a[/ilmath]
 This is not unique, if [ilmath]b\sim a[/ilmath] then we could write [ilmath][b][/ilmath] instead. (Equivalence classes are either equal or disjoint)
 Defined as [ilmath][a]:=\{b\in X\ \vert\ b\sim a\}[/ilmath]
 Often denoted [ilmath][a][/ilmath] for all the things equivalent to [ilmath]a[/ilmath]
 If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg [ilmath]\sim_\alpha[/ilmath] and [ilmath][\cdot]_\alpha[/ilmath]
 Sometimes different symbols are employed, for example [ilmath]\cong[/ilmath] denotes a topological homeomorphism (which is an equivalence relation on topological spaces)
See Also
 Relation
 Equivalence class
 Canonical projection of an equivalence relation
 Passing to the quotient  things are often factored through the canonical projection of an equivalence relation
 Equivalence relation induced by a map  while many things induce equivalence relations, the "purity" of any function doing so means this ought to be here
Notes
 ↑ This terminology means [ilmath]\sim \subseteq X\times X[/ilmath], as described on the relation page.
References

Old Page
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