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.

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: . Because the condition A1 ⊇ A2 ⊇ A3 ⊇ A4 ... we can deduce that the set is a subset of each set with . Thus,

.

Therefore, as is infinite, the intersection is infinite.

Now, if we consider the infinite intersection, i.e. the reasoning is slightly different. Take the sets

(this is, the open interval between 0 and .)

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

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

4 puzzles in 3 hours

step-by-step explanation: