Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Yesterday on my Abstract Algebra course, we were having a problem with equivalence relations. We had a given set: $$A = \{a, b, c\}$$

We found all the partitions of $A$, and one of them was: $$P = \{ \{a\} , \{b, c\} \}$$

Then we built an equivalence relation $S$ from this partition, where two elements are in equivalence relation if $a$ and $b$ belong to the same cell.

So the relation of equivalence is: $$S = \{ (a,a) , (b,b) , (c,c) , (b,c) , (c,b) \}$$

After this the professor, without explaining anything wrote:

The class of equivalence of $(b,c)$: $[(b,c)] = \{ (b,c) , (c,b) \}$

So can anyone explain this last line? Because I don't understand it.

share|cite|improve this question
Wouldn't it make more sense to say that the class of equivalence of $b$, $[b] = \{b,c\}$? The equivalence relation was defined on elements of $A$, not on pairs of elements from $A$. – Andrew Parker Feb 27 '12 at 20:38
well that's all the problem. I'm trying to figure it out and yet no clarity. I thought it like this too, but i don't know. – cprogcr Feb 27 '12 at 20:39
up vote 3 down vote accepted

When you have an equivalence relation $R$ on a set $X$, and an element $x\in X$, you can talk about the equivalence class of $x$ (relative to $R$), which is the set $$[x] = \{y\in X\mid (x,y)\in R\} = \{y\in X\mid (y,x)\in R\} = \{y\in X\mid (x,y),(y,x)\in R\}.$$

But I note that your professor did not say "equivalence class", he said "Class of Equivalence". That suggests he may be refering to some other (defined) concept. I would suggest that you talk to your professor directly and ask him what he means by "Class of Equivalence", and whether it is the same thing as "equivalence class"; explain what your understanding of "equivalence class" is, and why you would be confused if he said "The equivalence class of $(b,c)$ is $[(b,c)]={(b,c),(c,b)}$" (at least, I would be confused because in order to talk about "the equivalence class of $(b,c)$", I would need some equivalence relation defined on some set that contains $(b,c)$, and we don't seem to have that on hand).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.