Equivalence relation proof. If a ~ b then [a] = [b]

Question

I want to prove that if $a \sim b$ then $[a] = [b]$ (the set of a equals the set of b). I know that in order to prove this I must show that $[a] \subset [b]$ and $[b] \subset [a]$. How should the proof look?

Answer 1

For any $x\in[a]$ we have xRa and since aRb therefore because of equivalence we have xRb therefore $x\in[b]$ and $[a]\subset[b]$. With the same argument you can show the inverse

Answer 2

What this means is that if $x\in [a]$, then $x\in [b]$. In other words if $x\sim a$, then $x\sim b$ knowing that $a\sim b$. Can you take it from here?