[SOLVED] Real Analysis Proof: Intersection and Union of an Indexed Set
For each http://qaboard.cramster.com/Answer-B...2087503987.gif let http://qaboard.cramster.com/Answer-B...0833750886.gif. (Note Q is the set of all rationals).
(1) http://qaboard.cramster.com/Answer-B...2712509613.gif (that is the intersection over the entire index is the singleton set with 0).
and prove that
(2) http://qaboard.cramster.com/Answer-B...8025003118.gif (that is the union over the entire index is the set of all rationals in the interval [0,1) ).
Okay, wow. I am utterly stumped by this. Things I do know that 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:
= [0, 1/16)
= [0, 1/8)
= [0, 1/4)
= [0, 1/3)
= [0, 1/2)
So now I kind of 'see' better how the union would be everything including 0 but not reaching 1 because x (0, 1).
Any helps, hints, tips, and/or suggestions on getting started are greatly appreciated!
Thank you for your time.