Consider the following statements:
(I) The set \(\{\sin(1/x): x \in (0,1)\}\) is uncountable.
(II) The set \(\{(x,y) \in \mathbb{R}^2 : xy \in \mathbb{Z}\}\) is uncountable.
(III) The set of all \(2 \times 2\) real matrices with rational eigenvalues is uncountable.
Choose the correct answer: