In my last question The following groups are the same., three groups were given and I wanted to verify that they are isomorphic to each other. Derek suggested some points and I got my answer about one of them completely. But one was remained. I am trying to simplify my question about it again.
There; I had to show that $\frac{\mathbb R}{\mathbb Q}\cong\mathbb R$ since I need it. A theorem tells me that $\frac{\mathbb R}{\mathbb Q}$ is a vector space over $\mathbb Q$ because it is abelian divisible torsion-free and the same is true for $\mathbb R$. So I consider $\mathcal{B}$ be a basis for vector space $\mathbb R$ over $\mathbb Q$.
Now can I have $\mathcal{B}+\mathbb Q$ as a basis for vector space $\frac{\mathbb R}{\mathbb Q}$ over $\mathbb Q$ and concluding that $\#\mathcal{B}=\#(\mathcal{B}+\mathbb Q) $? I wanted to use what Derek pointed there for these two vector space again. Thanks.