Question

 Let a and b be distinct equivalence classes in a set S. Show that

a ∩ b = ∅

Answer 1

Solution:

Proof:

Lemma to be used: Given an equivalence relation R on set A, if a,b∈A then either [a]∩[b]=∅ or [a]=[b]

  

Now, suppose that $S_i\ and\ S_j$ are any two distinct equivalence classes of R. (We need to show that $S_i\ and\ S_j$ are disjoint.) Since $S_i\ and\ S_j$ are distinct, then $S_i\ne S_j$ . And since $S_i\ and\ S_j$ are equivalence classes of R, there must exist elements a and b in S such that $S_i = [a]\ and\ S_j = [b]$ . By above lemma, either [a] ∩ [b] = ∅ or [a] = [b].

But $[a] \ne [b]$ because $S_i\ne S_j$ . Hence [a] ∩ [b] = ∅