Take the 2-minute tour ×
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.

What is the standard notation for a set of equivalence relations? Specifically, I have a pair of objects, call them $x$ and $y$ and I denote the ordered pair as $\left(x,y\right)$. I have a set of equivalence relations such as:

$\left(x,y\right) \sim \left(x^{-1},y\right) \sim \left(x,y^{-1}\right) \sim \left(y^{-1},x^{-1}\right)$

I would like to write this compactly, but I'm unsure of what the standard notation would be. Is the following appropriate?

$\left\{\left(x,y\right),\left(x^{-1},y\right),\left(x,y^{-1}\right),\left(y^{-1},x^{-1}\right)\right\}$

share|improve this question
    
I think the usual way is to write just what you have written (perhaps followed by "$\forall x, y \in G$"), and say the quotient is $G/\sim$. –  Kundor Apr 23 '13 at 17:20

2 Answers 2

up vote 5 down vote accepted

You can represent an equivalence class by using a representative from the class, and denoting the entire class by, say, $[(a, b)]$: this represents the set of all ordered pairs $(x, y)$ such that $(x, y) \sim (a, b)$.

So, for a given equivalence relation denoted by $\sim$: one of its equivalence classes can be denoted: $$[(a, b)] = \{(x, y)\mid (x, y) \sim (a, b)\}$$

If there are many equivalence classes determined by an equivalence relation, and you want to denote the set of equivalence classes, you can list the equivalence classes as elements of a set:

  • $\{[(a, b)], [(c,d)], [(e, f)], \cdots [(y, z)]\}$ if there is a finite set of them.
  • For example, the set of equivalence classes determined by the equivalence relation of congruence modulo $4$ on the set of integers, you could write $\{[0], [1], [2], [3]\}$,
  • or, in the case of an infinite number of equivalence classes, like those corresponding to equality/identity on the natural numbers, one can write $\{[1], [2], [3], \cdots \}$.

If you are asking how to denote a set of different equivalence relations, where the elements of the set are relations, I'm not aware of the standard notation. (That's not to say it doesn't exist.)

share|improve this answer
    
What I want to do is have a way to denote the entire set of equivalence relations, like $P=\left\{\left(x,y\right),\left(x^{-1},y\right),\left(x,y^{-1}\right),\left(y^{‌​-1},x^{-1}\right)\right\}$, so that I can later define the quotient set as $G/P$, so I need a way to denote the entire set of equivlanece relations not just one of them. –  okj Apr 22 '13 at 19:27
    
Try using square brackets to denote that the elements of $P$ are equivalence classes:$$ P=\left\{\left[\left(x,y\right)\right],\left[\left(x^{-1},y\right)\right],\left[‌​\left(x,y^{-1}\right)\right],\left[\left(y^{‌​-1},x^{-1}\right)\right]\right\}$$ –  amWhy Apr 22 '13 at 21:38
    
@amWhy: Very nice answer +1 –  Amzoti Apr 23 '13 at 0:17

If $\sim$ is an equivalence class over $A$, then in many places we write $A/{\sim}$ as the set of equivalence classes.

This notation is similar, and on purpose, to the notation from algebra when writing $V/W$ for the quotient subspace, or $G/H$ for the quotient group, or $R/I$ when taking a quotient of a ring by an ideal.

The reason is that all those quotients actually induce an equivalence relation, and we have a natural structure on the set of equivalence classes.

share|improve this answer
    
I honestly don't know what could be wrong in this answer, and I would be very glad if someone would tell me. Perhaps something is wrong with the user posting it instead... –  Asaf Karagila Dec 16 '13 at 21:30
    

Your Answer

 
discard

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.