Confusion on why 2 equivalence classes are either equal or disjoint
Yes, the equivalence classes must be defined by the same equivalence relation. Otherwise the statement is not true. For example,take the set $X=\{1,2,3\}$ ande define these equivalence relations:
- for all $x,y\in X$, set $x$ and $y$ equivalent
- for all $x\in X$, $x$ is equivalent only with itself and with no other elements.
In the first case there is only one equivalence class, namely the whole $X$. In the second case there are three equivalence classes, $[1]$,$[2]$ and $[3]$. As you can see the classes of the second equivalence relation are not dijoint from the only class of the first equivalence relation.
Yes, the discussion is about one fixed equivalence relation $R$ on $X$, say.
If two classes $[x]_R$ and $[x']_R$ of $R$ intersect, $xRx'$ (transitivity via the intersection element) and so the classes are actually equal to each other.
We rarely mix equivalence relations; there is usually one under consideration, and we discuss its classes.