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

Answer: (B), (D)

G ∩ I is non-empty for every non-empty open interval I ⊆ ℝ

Answer 2

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).