Clarification for equivalence class

Question

I've been reading about equivalence class lately and I've a question.

In my book it's given, if there is a relation $R$ on set $\mathbb Z$ of integers, $$R=\{(a, b) : a, b \in \mathbb Z, a-b\text{ is divisible by }2\},$$ then this relation is an equivalence relation.

But now We can see that the relation has divided the set $\mathbb Z$ of integers into $2$ disjoint sets of $\mathbb E$ and $\mathbb O$ (even and odd). In $\mathbb E$ every element is related to $0$ and each other but not with the elements of $\mathbb O$ and in $\mathbb O$ every element is related to $1$ and each other but not with the elements of $\mathbb E$. Therefore $\mathbb E$ and $\mathbb O$ are forming an equivalence class and $\mathbb E$ can be written as $[0]$ and $\mathbb O$ can be written as $[1]$.

Now my question is since in both the sets each element of the set is related to other element of the set that means that every element in $\mathbb E$ and $\mathbb O$ is experiencing what $0$ in $\mathbb E$ and $1$ in $O$ is experiencing.

So why are we writing $0$ and $1$ in the brackets for representing the equivalence class? Is it because they are the smallest in value?

Answer 1

We write $[0]$ to signal that we're talking about the equivalence class of $0$, not the element $0\in\mathbb Z$.

And that $0$ and $1$ are used as the elements we use to denote the equivalence classes is just a choice the author has made. He could use a different even number every time he wanted to mention the equivalence class of even numbers, but it would probably be confusing to read, and definitely harder to write.

And as (equivalence) relations and thereby equivalence classes can be defined on any set, also those that don't consist of numbers, and can't even be ordered, "smallest in value" would make very little sense as a convention/rule for writing those.

Answer 2

$E$ can also written as $[2]$ or as $[4]$ or as $[-2].......$.

$O$ can also written as $[3]$ or as $[5]$ or as $[-1].......$.

Answer 3

You may read the symbol $[\cdot]$ as $\text{the equivalence class of}, $ so that $[0],$ for example would read the equivalence class of $0.$ The element $0$ is called a representative of the class. As others have pointed out, any element in the class may as well serve as its representative. But the main point is we're no longer thinking about individual elements, but the whole class as a single entity.