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