Help plz ! we say that 1) x\inuuu_(n\inN)A_n if x\inA_n for some n\inN and 2)x\innnn_(n\inN)A_n if x\inA_n for all n\inN prove or dispprove nnn_(n\inN)A_n\sube uuu_(n\inN)A_n for any index set N ?

1 Answer
Jun 21, 2018

True.

Explanation:

Let's call the intersection of all sets A and their union B.

We want to prove that A is a subset of B. By definition, this happens when every element of A is also an element of B, whereas the contrary is not necessarily true.

So, we need to show that

x \in A \implies x \in B

We only need to recall the definition, and everything gets really simple: if we assume that x \in A, it means that x \in A_n for all n \in N. On the other hand, in order to x \in B, it is sufficient that x \in A_n for at least one n \in N.

So, if we assume that x belongs to all the sets A_n, is it true that it belongs to at least one of the A_n? Well, of course it is.

On the other hand, if we know that x belongs to at least one of the A_n, can we conclude that it belongs to all the A_n? Not necessarily.

This is enough to prove that the set given by the intersection is a subset of the set given by the union.