Difference between revisions of "Equivalence relation"
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*[[Equivalence class]] | *[[Equivalence class]] | ||
**[[The equivalence classes of an equivalence relation partitions a set]]. | **[[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 | ||
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Contents
[hide]Definition
A relation, ∼, in X[Note 1] is an equivalence relation if it has the following properties[1]:
| Name | Definition | |
|---|---|---|
| 1 | Reflexive | ∀x∈X[(x,x)∈∼]. Which we write ∀x∈X[x∼x]. |
| 2 | Symmetric | ∀x,y∈X[M. Which we write ∀x,y∈X[x∼y⟹y∼x]. |
| 3 | Transitive | ∀x,y,z∈X[((x,y)∈∼∧(y,z)∈∼)⟹(x,z)∈∼]. Which we write ∀x,y,z∈X[(x∼y∧y∼z)⟹x∼z]. |
Terminology
- An equivalence class is the name given to the set of all things which are equivalent under a given equivalence relation.
- Often denoted [a] for all the things equivalent to a
- This is not unique, if b∼a then we could write [b] instead. (Equivalence classes are either equal or disjoint)
- Defined as [a]:={b∈X | b∼a}
- Often denoted [a] for all the things equivalent to a
- If there are multiple equivalence relations at play, we often use different letters to distinguish them, eg ∼α and [⋅]α
- Sometimes different symbols are employed, for example ≅ 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
References
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Old Page
An equivalence relation is a special kind of relation
Required properties
Given a relation R in A we require the following properties to define a relation (these are restated for convenience from the relation page)
Reflexive
A relation R if for all a∈A we have aRa
Symmetric
A relation R is symmetric if for all a,b∈A we have aRb⟹bRa
Transitive
A relation R is transitive if for all a,b,c∈A we have aRb and bRc⟹aRc
Definition
A relation R is an equivalence relation if it is:
- reflexive
- symmetric
- transitive
