(a) Show that if each of the sets \(S_{n}(n=1,2,3, \ldots)\) is countable, then the union \(S=\bigcup_{n=1}^{\infty} S_{n}\) is also countable.
(b) Show that if \(S\) and \(T\) are countable sets, then the Cartesian product \(S \times T\) is also countable. Hence show that \(\bigcup_{n=1}^{\infty} S^{n}\) is countable, where \(S^{n}=S \times S \times \cdots \times S\) ( \(n\) times).