Suppose $(S, ∼)$ is an equivalence relation and suppose $a, b ∈ S$. Show $[a] = [b]$ if $a ∼ b$ and $[a] ∩ [b] = ∅$ if $a \not\sim b$.

Question

I am a bit lost on this question to the point that I don't know where to start. I am confused as to how I am supposed to show this without a defined ~ relation. any help would be greatly appreciated, thank you.

Answer 1

If a ~ b, then for all x,
x in [a] iff x ~ a iff x ~ b iff x in [b].
Thus [a] = [b].

If not a ~ b and x in [a] $\cap$ [b], then
x ~ a and x ~ b, so a ~ b, a contradiction.
Thus [a] $\cap$ [b] is empty.