Proving union and intersection of indexed family of sets

Nov 2012
4
0
USA
I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].

I've asked the professor twice now for help and he hasn't been able to explain it at all.
 
Oct 2012
259
19
israel
show inclusions in both directions.
for the intersection :what x is between -n and 1/n for all n?
 

Plato

MHF Helper
Aug 2006
22,490
8,653
I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1).
And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0].
I think you are confused by the notation.
Let \(\displaystyle D_n = \left( { - n,\frac{1}{n}} \right)\).
Now consider two examples: \(\displaystyle D_3 \cup D_1 = \left( { - 3,1} \right)\;\& \;D_3 \cap D_1 = \left( { - 1,\frac{1}{3}} \right)\)

Now let's do an indexed example:
\(\displaystyle \bigcup\limits_{k = 1}^{100} {D_k } = \left( { - 100,1} \right)\;\& \;\bigcap\limits_{k = 1}^{100} {D_k } = \left( { - 1,\frac{1}{{100}}} \right)\)

Note how the right and left endpoints are limits.
Do you see how it works?
 
Nov 2012
4
0
USA
Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.
 

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Plato

MHF Helper
Aug 2006
22,490
8,653
Ok, I'm not familiar with latex so I can't get this into symbolic form but I'm going to attach a picture of what the question is. I have to prove it using the definition that for two things to be equal they each have to be a subset of the other.
Frankly I have no idea why you posted exactly what you posted exactly what I had already posted.
\(\displaystyle \lim _{n \to \infty } \bigcup\limits_{k = 1}^n {D_k } = \left( { - \infty ,1} \right)\;\& \,\;\;\lim _{n \to \infty } \bigcap\limits_{k = 1}^n {D_k } = \left( { - 1,0} \right]\)

BTW: Why not learn to use LaTeX?
 
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Nov 2012
4
0
USA
because I need a proof for the answer? not the answer itself? I know the answer but not the proof for it.
 
Nov 2012
4
0
USA
Also like I said I know from talking to the professor that the proof needs to use the definition of for things to be equal they have to be subsets of each other. I hope that is clear.

Also I might learn latex sometime but not tonight!
 
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Deveno

MHF Hall of Honor
Mar 2011
3,546
1,566
Tejas
try proving this "in-between" step:

for the first problem-

show that if k < m, that (Dk)U(Dm) = (-m,1/k). what is the smallest k can be, and what is the largest m can be?

for the second problem-

show that for k < m that (Dk)∩(Dm) = (-k,1/m). again: how small can k be, and how large can m be?

(the answer for k in both cases should be "easy". answering for m might take a little thought).

try drawing a picture with k = 3, and m = 4. draw another one with k = 1, and m = 10. what do you notice?