How to prove two equivalence classes are disjoint?

Question

I know how to prove when the two equivalence classes are not disjoint, i.e. $[a]=[b]$. I see that the proof works for proving a equivalence class is disjoint, but I don't get it. Can someone explain it to me?

Answer 1

Suppose we have two equivalence classes $[a]\not=[b]$. We'll show they are disjoint. Suppose $x\in[a]\cap[b]$. Then, $x\sim a$ and $x\sim b$. By symmetry, $a\sim x$, and hence, by transitivity, $a\sim b$. Therefore, $[a]=[b]$, which is a contradiction.