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?

Let S = {a + b√2 : a, b ∈ Z}, T₁ = {(-1+√2)ⁿ : n ∈ N} and T₂ = {(1+√2)ⁿ : n ∈ N}. Then which of the following statements is (are) TRUE? $S = \{a + b\sqrt{2} : a, b \in \mathbb{Z}\}$ $T_1 = \{(-1+\sqrt{2})^{n} : n \in \mathbb{N}\},\quad T_2 = \{(1+\sqrt{2})^{n} : n \in \mathbb{N}\}$ $\text{To find: which of (1), (2), (3), (4) are TRUE.}$ $\text{Plan: verify (1), (2), (3), (4) one by one.}$ $\text{Key: } (-1+\sqrt{2})(1+\sqrt{2}) = 1$ $(-1+\sqrt{2})(1+\sqrt{2}) = -1 + 2 = 1$ $\Rightarrow\ (-1+\sqrt{2})^{n} = (1+\sqrt{2})^{-n}$ $(-1+\sqrt{2}),\ (1+\sqrt{2}) \in S$ $S \text{ closed under } \times \Rightarrow (-1+\sqrt{2})^{n},\ (1+\sqrt{2})^{n} \in S$ $(1)\ \mathbb{Z} \subset S,\ T_1 \subset S,\ T_2 \subset S$ $\Rightarrow \mathbb{Z} \cup T_1 \cup T_2 \subset S\ \ \checkmark$ $t_n = (\sqrt{2}-1)^{n},\ 0 < \sqrt{2}-1 < 1$ $t_n \downarrow 0 \text{ as } n \to \infty$ $(\sqrt{2}-1)^8 \approx 8.66\times 10^{-4} > \dfrac{1}{2024}$ $(\sqrt{2}-1)^9 \approx 3.59\times 10^{-4} < \dfrac{1}{2024}$ $(\sqrt{2}-1)^9 \in T_1 \cap \left(0,\dfrac{1}{2024}\right)$ $(2)\ \text{FALSE}\ \times$ $u_n = (1+\sqrt{2})^{n},\ 1+\sqrt{2} > 1$ $u_n \uparrow \infty \text{ as } n \to \infty$ $(1+\sqrt{2})^8 \approx 1154 < 2024$ $(1+\sqrt{2})^9 \approx 2786 > 2024$ $(1+\sqrt{2})^9 \in T_2 \cap (2024,\infty) \neq \phi$ $(3)\ \text{TRUE}\ \checkmark$ $\cos(\pi(a+b\sqrt{2})) + i\sin(\pi(a+b\sqrt{2})) = e^{i\pi(a+b\sqrt{2})}$ $= e^{i\pi a}\cdot e^{i\pi b\sqrt{2}} = (-1)^{a}\left(\cos(\pi b\sqrt{2}) + i\sin(\pi b\sqrt{2})\right)$ $\text{Real integer requires } \sin(\pi b\sqrt{2}) = 0$ $\Rightarrow b\sqrt{2} = k \in \mathbb{Z} \Rightarrow b = 0\ (\sqrt{2}\ \text{irrational})$ $b = 0 \Rightarrow (-1)^{a} \in \mathbb{Z}$ $(4)\ \text{TRUE}\ \checkmark$ $\boxed{(1),\ (3),\ (4)\ \text{are TRUE}}$