Invariant of an equivalence relation

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Note: see invariant for other uses of the term.
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Let [ilmath]S[/ilmath] be a set and let [ilmath]\sim\subseteq S\times S[/ilmath] be an equivalence relation[Note 1] on [ilmath]S[/ilmath], let [ilmath]W[/ilmath][Note 2] be any set and let [ilmath]f:S\rightarrow W[/ilmath] be any function from [ilmath]S[/ilmath] to [ilmath]W[/ilmath]. Then[1]:

  • We say "[ilmath]f[/ilmath] is an invariant of [ilmath]\sim[/ilmath]" if[Note 3]:
    • [ilmath]\forall a,b\in S[a\sim b\implies f(a)=f(b)][/ilmath] - in other words, [ilmath]f[/ilmath] is constant on the equivalence classes of [ilmath]\sim[/ilmath].

Complete invariant

With the setup of [ilmath]S[/ilmath], [ilmath]W[/ilmath], [ilmath]\sim[/ilmath] and [ilmath]f:S\rightarrow W[/ilmath] as above define a "complete invariant" as follows[1]:

  • "[ilmath]f[/ilmath] is a complete invariant of [ilmath]\sim[/ilmath]" if[Note 3]:


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

  • "[ilmath]f[/ilmath] is an invariant of [ilmath]\sim[/ilmath]"[1]
  • "[ilmath]\sim[/ilmath] is invariant under [ilmath]f[/ilmath]"
    • This makes sense as we're saying the [ilmath]a\sim b[/ilmath] property holds (doesn't vary) "under" (think "image of [ilmath]A[/ilmath] under [ilmath]f[/ilmath]"-like terminology) [ilmath]f[/ilmath], that [ilmath]f(a)=f(b)[/ilmath]
  • "[ilmath]\sim[/ilmath] invariance of [ilmath]f[/ilmath]"
    • This works better when the relations have names, eg "equality invariance of Alec's heuristic" (that's a made up example) and this would be a proposition or a claim.

Examples and instances

TODO: Create a category and start collecting

See also


  1. keep in mind that equality is itself an equivalence relation
  2. Think of [ilmath]W[/ilmath] as [ilmath]W\text{hatever} [/ilmath] - as usual (except in Linear Algebra where [ilmath]W[/ilmath] is quite often used for vector spaces
  3. 3.0 3.1 See "definitions and iff"


  1. 1.0 1.1 1.2 Advanced Linear Algebra - Steven Roman
Grade: A
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More on the fundamentals of mathematics would be good