Exercise 819Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

$\mathbb{Q}$ is dense but countable

#rationals#density#countability

Statement

Explain why $\mathbb{Q}$ is countable yet dense in $\mathbb{R}$ .

Hint

Density means $\mathbb{Q}$ meets every nonempty open interval; countability means $|\mathbb{Q}|=\aleph_0$ .

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Countability. Shown earlier: $|\mathbb{Q}|=\aleph_0$ .

Density. For any $a<b$ , some rational $p/q$ lies in $(a,b)$ (choose $q$ large enough).

Contrast. Density and countability coexist; "most" reals are irrational (uncountable), but every interval contains rationals.

Conclusion. $\mathbb{Q}$ is a countable dense subset of $\mathbb{R}$ .