I have the following problem:
Let R be a set with an equivalence relation ∼, prove that for all x, y ∈ X, either [x] = [y] or [x] ∩ [y] = ∅. (Here [x] denotes the equivalence class of an element x ∈ R).
I suppose that [x] ∩ [y] $\neq $ ∅, then there exists z $\in$[x]$\cap$[y], then zRx and zRy by definition ([x]={z$\in$R|z~x} and [y]= {z$\in$R|z~y}).
By symmetry we get xRz, with zRy and transitivity we get xRy.
So [x]$\subseteq$[y] and by symmetry we get [y]$\subseteq$[x] which implies that [x]=[y] else my assumption was false, and in that case [x] ∩ [y] = ∅.
Therefore, either [x] = [y] or [x] ∩ [y] = ∅.
Is this correct? I would appreciate any help and/or feedback.