Question

Decide which of the following represent true statements about the nature of set. For any that are false, provide a specific example where the statement in question does not hold.

(a) If A1 ⊇ A2 ⊇ A3 ⊇ A4 ... are all sets containing an infinite number of elements, then the intersection n-1 An is infinite as well.

Answer 1

If the intersection is finite the statement is true, but if the intersection is infinite the statement is false.

Explanation:

From the statement of the problem I am not sure if the intersection is finite or infinite. Then, I will study both cases.

Let us consider first the finite case:
`A = \cap_(i=1)^(n)A_i`. Because the condition A1 ⊇ A2 ⊇ A3 ⊇ A4 ... we can deduce that the set
`A_n` is a subset of each set
`A_i` with
`i\leq n`. Thus,

`\cap_(i=1)^(n)A_i = A_n`.

Therefore, as
`A_n` is infinite, the intersection is infinite.

Now, if we consider the infinite intersection, i.e.
`A = \cap_(k=1)^(\infty)A_k` the reasoning is slightly different. Take the sets

`A_k = (0,1/k)` (this is, the open interval between 0 and
`1/k`.)

Notice that (0,1) ⊇ (0,1/2) ⊇ (0, 1/3) ⊇(0,1/4) ⊇...So, the hypothesis of the problem are fulfilled. But,

`\cap_(k=1)^(\infty)(0,1/k) = \empyset`

In order to prove the above statement, choose a real number
`x` between 0 and 1. Notice that, no matter how small
`x` is, there is a natural number
`K` such that
`1/K<x`. Then, the number
`x` is not in any interval
`(0,1/k)` with
`k>K`. Therefore,
`x` is not in the set [tex]\cap_{k=1}^{\infty}(0,1/k)[\tex].