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

Answer:

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

Step-by-step 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 $1/K$. 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].