Prove that $[a]=[b]$ iff $a\sim b$.

Question

If $[a]=[b]$ then $a$ is in $[b]$ and $b$ is in $[a]$. If $a \sim b$, then $[a]$ is a subset of $[b]$ and $[b]$ is a subset of $[a]$. Then using the equivalence properties and showing that there is a related element between $a$ and $b$, $[a]=[b]$. Am I right?

Answer 1

Perhaps it will help to write out what the sets $[a]$ and $[b]$ are:

$$[a]=\{x\mid a\sim x\}\\ [b]=\{x\mid b\sim x\}.$$

Obviously, $a\in [a]$ and $b\in [b]$. Now,

$[\implies]$ We have $$[a]=[b]\implies a\in [b]\implies a\sim b.$$

$[\impliedby]$ Suppose $a\sim b$. Then $$a\sim x \iff x\sim b \text{ (by transitivity).}$$ But $x\sim a \iff x\in [a] $ and likewise for $[b]$, so we have $$x\in[a] \iff x\in [b]\equiv [a]=[b].\square$$

Answer 2

Let $c\in [a]$. Then $c\mathcal{R}a$. Since $a\mathcal{R}b$ and $\mathcal{R}$ is an equivalence relation, we have $c\mathcal{R}b$. Hence, $c\in[b]$, i.e. $[a]\subseteq [b]$. By the same argument, you obtain $[b]\subseteq [a]$, which means that $[a]=[b]$.