For each let . (Note Q is the set of all rationals).

Prove that:

(1) (that is the intersection over the entire index is the singleton set with 0).

and prove that

(2) (that is the union over the entire index is the set of all rationals in the interval [0,1) ).

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Okay, wow. I am utterly stumped by this. Things I do know that $\displaystyle E_x$ is the set of all sets with

the form [0 , x) for all x in the interval (0, 1). The intersection is what all of those sets have in common, and I can see that it is 0; but I can't figure out the proof. Since it is proving two sets equal, I would assume having to "chase elements" and prove the set on the right side is a subset of the left and vice versa. We did a proof similar to this in class, and our professor proved it using contradiction. So then maybe I will also need to use contradiction to prove these two statements?

As for the union proof, I'm also thinking of a chasing elements proof. But again, where to start is what I am struggling with.

I listed out some sets like:

$\displaystyle E_{1/16}$ = [0, 1/16)

$\displaystyle E_{1/8}$ = [0, 1/8)

$\displaystyle E_{1/4}$ = [0, 1/4)

$\displaystyle E_{1/3}$ = [0, 1/3)

$\displaystyle E_{1/2}$ = [0, 1/2)

So now I kind of 'see' better how the union would be everything including 0 but not reaching 1 because x $\displaystyle \in$ (0, 1).

Any helps, hints, tips, and/or suggestions on getting started are greatly appreciated!

Thank you for your time.