Let $V$ be a vector space over $\mathbb{R}$ and $W$ be a subspace of $V$. We know that the congruence module $W$ is an equivalence relation and that the quotient set $V/W$ is composed of equivalence classes. Now if $W=\left \{ 0 \right \}$, then $\left [ v \right ]=\left \{ x\in V;x\equiv v,\textrm{mod}\: 0 \right \}=\left \{ v \right \}$, then $V/W=\left \{ \left \{ v \right \};v\in V \right \}$.
But I saw in a book that this result is written as $V/W=V$.
What is the reason to write that way?
The elements of the quotient set are not sets?
In my opinion this way of writing is incorrect.
Thanks for your help.
You are correct; the sets are not equal but isomorphic, so it should be $V/W \cong V$.