Exercise 824Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

$\mathbb{Q}$ is countable

#rationals#countability#surjection

Statement

Show that $\mathbb{Q}$ is countably infinite.

Hint

$\mathbb{Q}$ is the image of $\mathbb{Z}\times\mathbb{N}^*$ under $(p,q)\mapsto p/q$ , a countable set.

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Solution

Surjection. $(p,q)\mapsto p/q$ is a surjection $\mathbb{Z}\times\mathbb{N}^*\to\mathbb{Q}$ . Both factors are countable, so $\mathbb{Z}\times\mathbb{N}^*$ is countable. Its image is $\mathbb{Q}$ .

Infinite. $\mathbb{Q}$ contains $\mathbb{N}$ , so is infinite.

Countable + infinite. $\mathbb{Q}$ is countably infinite.

Conclusion. $|\mathbb{Q}|=\aleph_0$ .

$\mathbb{Q}$ is countable

Statement

Solution

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