Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.
I was given the following question as an assignment and I have proved that $\sim$ is an equivalence relation on $G$, that was not difficult. The second part of the question asks us to show that the equivalence classes of $~$ are left cosets of $H$ in $G$. I know that a left coset is $gH = \{gh : h \in H\}$ what must I show prove that the equivalence classes of $\sim$ are left cosets?
I think the boils down to more of a question about proving equivalence classes more than anything. So my real question is how do you prove or show equivalence classes of a relation? I think if I was clear on that the question will be much easier.
Your job, if you choose to accept it, is to take any two elements $x,y \in G$ and to prove that $x \sim y$ if and only if there exists $g \in G$ such that $x \in gH$ and $y \in gH$.