Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 8E / b / ii / Solution

Yes. The set ${\mathbb{Q}}^2$ is countable, hence the set of unordered pairs of distinct rational points is countable. Two distinct points determine at most one line. Mapping each such pair to its line therefore has countable image, and every line in the stated collection lies in that image. Thus the collection is countable.

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