Difference between revisions of "Invariant of an equivalence relation"

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(Created page with "{{Stub page|grade=A|msg=Find more references and flesh out}} ==Definition== Let {{M|S}} be a set and let {{M|\sim\subseteq S\times S}} be an equivalence relation on {{...")
 
m (Added link to page that will be disambiguation, added comment to remind the reader that equality is itself an equivalence relation)
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: '''Note: ''' see [[invariant]] for other uses of the term.
 
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==Definition==
 
==Definition==
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Let {{M|S}} be a [[set]] and let {{M|\sim\subseteq S\times S}} be an [[equivalence relation]] on {{M|S}}, let {{M|W}}<ref group="Note">Think of {{M|W}} as {{M|W\text{hatever} }} - as usual (except in [[Linear Algebra (subject)|Linear Algebra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. Then{{rALASR}}:
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Let {{M|S}} be a [[set]] and let {{M|\sim\subseteq S\times S}} be an [[equivalence relation]]<ref group="Note">keep in mind that [[equality]] is itself an equivalence relation</ref> on {{M|S}}, let {{M|W}}<ref group="Note">Think of {{M|W}} as {{M|W\text{hatever} }} - as usual (except in [[Linear Algebra (subject)|Linear Algebra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. Then{{rALASR}}:
 
* We say "{{M|f}}'' is an invariant of ''{{M|\sim}}" if<ref group="Note" name="definition">See "''[[definitions and iff]]''"</ref>:
 
* We say "{{M|f}}'' is an invariant of ''{{M|\sim}}" if<ref group="Note" name="definition">See "''[[definitions and iff]]''"</ref>:
 
** {{M|1=\forall a,b\in S[a\sim b\implies f(a)=f(b)]}} - in other words, {{M|f}} is [[constant on]] the [[equivalence classes]] of {{M|\sim}}.
 
** {{M|1=\forall a,b\in S[a\sim b\implies f(a)=f(b)]}} - in other words, {{M|f}} is [[constant on]] the [[equivalence classes]] of {{M|\sim}}.

Revision as of 18:51, 9 November 2016

Note: see invariant for other uses of the term.
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Definition

Let S be a set and let ∼⊆S×S be an equivalence relation[Note 1] on S, let W[Note 2] be any set and let f:S→W be any function from S to W. Then[1]:

  • We say "f is an invariant of ∼" if[Note 3]:

Complete invariant

With the setup of S, W, ∼ and f:S→W as above define a "complete invariant" as follows[1]:

Terminology

It's hard to be formal in English, however we may say any of the following:

  • "f is an invariant of ∼"[1]
  • "∼ is invariant under f"
    • This makes sense as we're saying the a∼b property holds (doesn't vary) "under" (think "image of A under f"-like terminology) f, that f(a)=f(b)

Examples and instances

TODO: Create a category and start collecting

See also

Notes

  1. Jump up ↑ keep in mind that equality is itself an equivalence relation
  2. Jump up ↑ Think of W as Whatever - as usual (except in Linear Algebra where W is quite often used for vector spaces
  3. ↑ Jump up to: 3.0 3.1 See "definitions and iff"

References

  1. ↑ Jump up to: 1.0 1.1 1.2 Advanced Linear Algebra - Steven Roman
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