Is $X/{\sim}:=\{[x]: x\in X\}=\emptyset$ or $X/{\sim}:=\{[x]: x\in X\}=\{\emptyset\}$ when $X=\emptyset$?

Question

Let $X:=\emptyset$.
Suppose $R\subset X\times X$.
Then, $R=\emptyset$.
$R$ is a relation (the only relation) on $X$.
$R$ is an equivalence relation.
Let ${\sim} := R$.
Let $[x]:=\{y\in X:y\sim x\}$ for any $x\in X$.

1. $[x]=\{y\in X:y\sim x\}=\emptyset$ for any $x\in X$ since there is no $y$ such that $y\in X$.
   Therefore $X/{\sim}:=\{[x]: x\in X\}=\{\emptyset\}$.
2. $X/{\sim}:=\{[x]: x\in X\}=\emptyset$ since $X=\emptyset$.

Which is right?

The author of the book which I am reading now says $X/{\sim}:=\{[x]: x\in X\}=\emptyset$ since $X=\emptyset$.
And I wonder the author is right.

I want to know the reason why 1 is wrong and why 2 is right.

Answer 1

The meaning of $X/\!\sim\,=\{[x]:x\in X\}$ is that each element of $X/\!\sim$ is $[x]$ for at least one element $x\in X$. Since $X=\varnothing$ has no elements, there is no such $x$ and therefore no such $[x]$. So $X/\!\sim$ has no elements; that is, $X/\!\sim=\varnothing$.

Answer 2

$X/\sim$ is a set of disjoint subsets of $X$. And since $X$ has no subsets, $X/\sim$ is the empty set.