Not sure if this is a suitable question here, but I'm having trouble understanding the intuition behind a theorem I've read in a textbook. So it says the following:
"If $\mathscr R$ is an equivalence relation on a set $A$, then every element $a \in A$ is in exactly one equivalence class. In particular, $a \mathscr R b$ if, and only if, $[a]=[b]$."
I'm having trouble how showing that the "if and only if" statement holding implies that $a$ is in a unique equivalence class. Like I understand proving the "if and only if" part itself, but I don't really know the intuition behind it. I'm guessing it sort of means that either every two equivalence classes are equal or they are disjoint. Could someone please elaborate?
Thanks in advance.