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Consider the set S = {a + b√2: a, b ∈ Z}, and the sets T₁ = {⌊(−1 + √2)ⁿ⌋: n ∈ N} and T₂ = {⌊(1 + √2)ⁿ⌋: n ∈ N}. Which of the following statements is/are correct?
- The union of Z, T₁, and T₂ is a subset of S.
- The intersection of T₁ with {0, 1/2024} is the empty set.
- The set T₂ has elements greater than 2024.
- For any integers a and b, the expression cos(π(a + b√2)) + i sin(π(a + b√2)) belongs to the integers if and only if b equals 0, where i is the imaginary unit.
Correct answer: The union of Z, T₁, and T₂ is a subset of S.
Solution
The union of Z, T₁, and T₂ is a subset of S because the elements of T₁ and T₂ are of the form ⌊(−1 + √2)ⁿ⌋ and ⌊(1 + √2)ⁿ⌋, which can be expressed as a + b√2, where a and b are integers.
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