Question

Let {Aₙ} ₙ₌₁ be a family of sets, where Aₙ=[-2n+1),1/2n). Which of the following statements are true?

Answer 1

The intersection of any two sets Aₙ and Aₙ₊₁ is the empty set. The set Aₙ is a subset of Aₙ₊₁ for all values of n.

Step-by-step explanation:

The given family of sets is defined as:

Aₙ = [-2n+1, 1/2n)

To determine which statements are true, let's analyze each option:

1. Statement 1: The intersection of any two sets Aₙ and Aₙ₊₁ is the empty set.
2. This statement is true because for any n, the interval [-2n+1, 1/2n) does not overlap with the interval [-2(n+1)+1, 1/2(n+1)), resulting in an empty intersection.
3. Statement 2: The union of any two sets Aₙ and Aₙ₊₁ is the set Aₙ.
4. This statement is false because the union of Aₙ and Aₙ₊₁ would include the intervals [-2n+1, 1/2n) and [-2(n+1)+1, 1/2(n+1)), resulting in a larger interval.
5. Statement 3: The set Aₙ is a subset of Aₙ₊₁ for all values of n.
6. This statement is true because the interval [-2n+1, 1/2n) is entirely contained within the interval [-2(n+1)+1, 1/2(n+1)).

Therefore, the correct statements are:

• The intersection of any two sets Aₙ and Aₙ₊₁ is the empty set.
• The set Aₙ is a subset of Aₙ₊₁ for all values of n.