Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$

Question

To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$

Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$. Hence they are not isomorphic to each other. Is that right, or do I need to prove something else?

Thanks.

Answer 1

In $\mathbb Q$ given $x$ and $y$ there are always non-zero integers $m$ and $n$ so that $mx=ny$.

In $\mathbb R$ this is not always the case, consider $x=\sqrt{2}$ and $y=1$

Answer 2

Recall that an isomorphism is a bijective homormophism between these two groups. Since bijections preserve the cardinality of a set, and $\Bbb{Q}$ and $\Bbb{R}$ do not have the same cardinality, there is no bijection from one to the other.

Answer 3

By definition any isomorphism would be a bijection between the underlying sets $\mathbb{Q}$ and $\mathbb{R}$, but those sets don't even have the same cardinality.