Disjoint Equivalence Why do equivalence classes, on a particular set, have to be disjoint? What's the intuition behind it? I'd appreciate your help
Thank you!

The idea behind an equivalence relation is to generalize the notion of equality.
The idea behind the equality relation is that something is only equal to itself. So two distinct objects are not equal.
With equivalence relation, if so, we allow two things to be "almost equal" (namely equal where it count, and we don't care about their other distinctive properties). So the equivalence class of an object is the class of things which are "almost equal" to it. Clearly if $x$ and $y$ are almost equal they have to have the same class of almost equal objects; and similarly if they are not almost equal then it is impossible to have an object almost equal to both.
