Prove that two equivalence classes of two elements are equal iff the elements are related

Question

Show that for equivalence relation $\sim$ on the set $A$ and $a,b\in A,[a]=[b] \iff a\sim b$.

This is an advanced practice problem.

Answer 1

Recall that an equivalence relation is reflexive, symmetric and transitive.

By reflexivity $a\in[a]$ and $b\in[b]$. If $[a]=[b]$ then $b\in[a]$ as well, and by definition of $[a]$ we have $a\sim b$.

If $a\sim b$, we need to show a double-sided inclusion, let $c\in[b]$ then $b\sim c$, and by transitivity $a\sim c$ and therefore $c\in[a]$, so $[b]\subseteq[a]$.

I am leaving you the final case of showing $[a]\subseteq[b]$ and deducing the equality.

Answer 2

Note also that equivalence classes partition $A$, and so for $a,b\in A,[a] \neq[b] \iff [a] \cap [b] = \emptyset$, or equivalently $ [a] = [b] \iff [a]\cap [b] \neq \emptyset$.