let $H,K \subseteq G$ all be groups. A relation ~ on $G$ is defined by x ~ y $\iff x = hyk$ for some $h\in H$ and $k\in K$. I'm to prove that ~ is an equivalence relation on $G$.
I was just wondering if when I'm to prove reflexivity if this holds:
x~x:
$x = hxk$
$h^{-1}xk^{-1} = h^{-1}hxkk^{-1} = x $ , since $h^{-1} \in H$ and $k^{-1} \in K$ and so it's proven?
Ie it doesn't have to be the same element of $H$ and the same element of $K$ right?