Difference between revisions of "Equivalence relation"
From Maths
m (→See Also: Linking out to new pages) |
m (Fixing mathematical notation typo) |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 17: | Line 17: | ||
! 2 | ! 2 | ||
| [[Relation#Types_of_relation|Symmetric]] | | [[Relation#Types_of_relation|Symmetric]] | ||
| − | | {{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 | ||
| Line 34: | Line 34: | ||
*[[Relation]] | *[[Relation]] | ||
*[[Equivalence class]] | *[[Equivalence class]] | ||
| − | **[[ | + | **[[The equivalence classes of an equivalence relation partitions a set]]. |
| + | * [[Canonical projection of an equivalence relation]] | ||
* {{link|Passing to the quotient|function}} - things are often factored through the [[canonical projection of an equivalence relation]] | * {{link|Passing to the quotient|function}} - 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 | * [[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 | ||
| − | ** [[Factoring a function through the | + | ** [[Factoring a function through the projection of an equivalence relation induced by that function yields an injection]] |
| + | *** [[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]] | ||
==Notes== | ==Notes== | ||
Latest revision as of 15:18, 12 February 2019
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
This page is waiting for a final review, then this notice will be removed.
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
