Question

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` ?

Answer 1

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.