Question

Consider
\(𝐺 = \left\{𝑚 + 𝑛\sqrt2\ ∶\ 𝑚, 𝑛 ∈ \Z\right\}\)
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE ?

• A. G is a cyclic subgroup of ℝ under addition
• B. G ∩ I is non-empty for every non-empty open interval I ⊆ ℝ
• C. G is a closed subset of ℝ
• D. G is isomorphic to the group ℤ × ℤ, where the group operation in ℤ × ℤ is defined by (m₁, n₁) + (m₂, n₂) = (m₁ + m₂, n₁ + n₂)

Answer 1

The Correct Option is B, D

(A) is false. \( G \) is not cyclic because it is not generated by any single element, given the presence of \( \sqrt{2} \) in its structure. - (B) is true. The set \( G \) is dense in \( \mathbb{R} \), meaning it intersects every non-empty open interval in \( \mathbb{R} \). - (C) is false. \( G \) is not closed because there are elements in \( \mathbb{R} \) that cannot be approximated by elements of \( G \), given its form involving \( \sqrt{2} \). - (D) is true. The structure of \( G \) is isomorphic to \( \mathbb{Z} \times \mathbb{Z} \) because it is essentially the set of integer pairs where each element in \( G \) corresponds to an ordered pair \( (m,n) \) in \( \mathbb{Z} \). Thus, the correct answers are (B) and (D).