Exercise 1.6.5 (The additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic)

Question

Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.

Richard Ganaye · Accepted Answer

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

\begin{proof} 
$\Q$ has the same cardinality as $\N =\Z^+$, and $\R$ has not the same cardinality as $\N$ ($|\Q| = \aleph_0,\ \ |\R| > \aleph_0$), so there is no bijection from $\R$ to $\Q$. A fortiori, there is no isomorphism from $\R$ to $\Q$.
\end{proof}

Exercise 1.6.5 (The additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic)

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Exercise 1.6.5 (The additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic)

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