Question

Prove or disprove: The set \(\mathbb{Q}^{100}\) is countably infinite.

Short answer

Yes, the set \(\mathbb{Q}^{100}\) is countably infinite.

Step-by-step solution

Definition of Countable Infinite Set

A Countable Infinite Set is a set which has the same cardinality (size) as the set of natural numbers (\(\mathbb{N}\)). This means we can create a bijective function that maps every member of our set to a unique natural number.

Recalling the example of countability for \(\mathbb{Q}\)

For any one-dimensional rational number set (i.e., \(\mathbb{Q}\)), we already know this is countable. We can list all pairs of integers such that the first is not zero, and every possible rational number will appear somewhere on this list.

Argument for countability in higher dimensions

We can extend this methodology to \(\mathbb{Q}^{100}\). We just now have to list 100-tuples instead of tuples. We can imagine a countably infinite lists of countably infinite rows (one for each dimension). When we think of traversing this structure in a diagonal manner, each rational 100-tuple corresponds to a unique natural number, and this shows a bijective mapping between \(\mathbb{Q}^{100}\) and \(\mathbb{N}\). Therefore, \(\mathbb{Q}^{100}\) is a countably infinite set.

Conclusion

The set \(\mathbb{Q}^{100}\) is countably infinite because there exists a way to map all elements of the set to a unique natural number, by extending the diagonal argument used for the one-dimensional case.