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

The correct option is (B) : G ∩ I is non-empty for every non-empty open interval I ⊆ ℝ and (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).