Showing that $Y\cong W$ but $X/Y\not \cong X/W$. I was trying to solve the following question:

Let $X=\mathbb{Z}_4\times\mathbb{Z}_2$, $Y=\{0,2\}\times\{0\}$ and $W=\{0\}\times \mathbb{Z}_2$. Show that $Y\cong W$ but $X/Y\not \cong X/W$.

I guess I must verify manually the Isomorphism theorems (link), but how should I do it without those theorems? I don't understand how to verify the statement.

Hint:  Show that $Y\cong W\cong\Bbb Z_2$.
Secondly,  $X/Y\cong V_4=\Bbb Z_2×\Bbb Z_2$, but $X/W\cong\Bbb Z_4$.
