Let \( S = \{ a + b\sqrt{2} : a, b \in \mathbb{Z} \} \), \( T_1 = \{ (-1 + \sqrt{2})^n : n \in \mathbb{N} \} \), and \( T_2 = \{ (1 + \sqrt{2})^n : n \in \mathbb{N} \} \). Then which of the following statements is (are) TRUE?
\( \mathbb{Z} \cup T_1 \cup T_2 \subset S \)
\( T_1 \cap \left(0, \frac{1}{2024}\right) = \phi \), where \( \phi \) denotes the empty set
\( T_2 \cap (2024, \infty) \neq \phi \)
For any given \( a, b \in \mathbb{Z} \), \( \cos\!\big(\pi(a + b\sqrt{2})\big) + i \sin\!\big(\pi(a + b\sqrt{2})\big) \in \mathbb{Z} \) if and only if \( b = 0 \) where \( i = \sqrt{-1} \)
