I am reading "Sets, Numbers and Topology" by Masahiko Saito.
Let $R$ be an equivalence relation on a set $A$.
If we collect all mutually $R$-equivalnent elements in $A$, we get a subset of $A$.
Let $\mathcal{F}$ be the set of all such subsets of $A$.
Then, $\mathcal{F}$ is a classification of $A$.
An element of $\mathcal{F}$ is called an equivalence class with respect to $R$.
What does "collect all mutually $R$-equivalnent elements in $A$" mean?
I cannot understand this operation exactly.
The phrase all mutually $R$-equivalent elements is misleading. It's like specifying "all integers which end with the same digit"—which is obviously only defined once we choose the digit.
What it's really saying is this.
Choose some element $a$ of $A$. (Crucial but not mentioned.) Then $R$ defines a subset $S\subset A$ each of whose elements is equivalent to $a$: that is, every $x\in S$ satisfies $xRa$.
The mutually equivalent part is that for any $x, y\in S$, each of $x$ and $y$ is equivalent to the other: $xRy$ and $yRx$. Every element of $S$ is equivalent, under $R$, to every other.
This follows from the definition of an equivalence relation, by which $R$ is both transitive (if $aRbRc$ then $aRc$) and commutative (if $aRb$ then $bRa$).
The book goes on to say that $S$ is called an equivalence class, and that the set of equivalence classes is called a classification.