"Definition 6. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. For each $x \in X$, we define
$x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$
which is called the equivalence class determined by the element $x$.
The set of all such equivalence classes on $X$ is denoted by $X/\mathscr E$; that is, $X/\mathscr E=\{x/\mathscr E \mid x \in X\}$. The symbol $X/\mathscr E$ is read "$X$ modulo $\mathscr E$," or simply "$X$ mod $\mathscr E$".
Source: Set Theory by You-Feng Lin, Shwu Yeng T. Lin
In the definition, I don't understand "For each $x \in X$, we define
$x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$
which is called the equivalence class determined by the element x."
"$x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$" means x/$\mathscr E$ has y in X as its element and $y\mathscr Ex \Leftrightarrow (y, x) \in \mathscr E$. So shouldn't it be 'For each $y \in X$', rather than "For each $x \in X$'"?
He just means that for each $x \in X$ there is an equivalence class associated with that element, called $x/\mathscr{E}$. Within the definition of $x/\mathscr{E}$, yes you do look at for all $y \in X$, but that is not what he was talking about when he wrote "for each $x \in X$" earlier.