What is the standard notation for a set of equivalence classes?

Question

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?

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

Answer 1

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 $\{[1], [2], [3], \cdots \}$.

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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.)

Answer 2

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.