Answer
See explanation
Work Step by Step
A straightforward way to prove \(3\mathbb{Z}\) is countable is to use the fact (from Exercise 3) that \(3\mathbb{Z}\) has the same cardinality as \(\mathbb{Z}\). Since \(\mathbb{Z}\) is known to be countable, any set in bijection with it must also be countable. Formally:
1. **From Exercise 3**: We have a bijection \(f \colon \mathbb{Z} \to 3\mathbb{Z}\) given by \(f(n) = 3n\).
2. **\(\mathbb{Z}\) is countable**: By standard results, the set of integers is countable.
3. **Bijections preserve countability**: If \(A\) is countable and \(B\) is in bijection with \(A\), then \(B\) is countable as well.
Hence, \(3\mathbb{Z}\) is countable.
