# Proving union and intersection of indexed family of sets

• Nov 4th 2012, 01:10 PM
skippenmydesign
Proving union and intersection of indexed family of sets
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.
• Nov 4th 2012, 01:40 PM
hedi
Re: Proving union and intersection of indexed family of sets
show inclusions in both directions.
for the intersection :what x is between -n and 1/n for all n?
• Nov 4th 2012, 01:54 PM
Plato
Re: Proving union and intersection of indexed family of sets
Quote:

Originally Posted by skippenmydesign
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 4th 2012, 04:56 PM
skippenmydesign
Re: Proving union and intersection of indexed family of sets
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.
• Nov 4th 2012, 05:12 PM
Plato
Re: Proving union and intersection of indexed family of sets
Quote:

Originally Posted by skippenmydesign
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?
• Nov 4th 2012, 05:14 PM
skippenmydesign
Re: Proving union and intersection of indexed family of sets
because I need a proof for the answer? not the answer itself? I know the answer but not the proof for it.
• Nov 4th 2012, 05:28 PM
skippenmydesign
Re: Proving union and intersection of indexed family of sets
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!
• Nov 4th 2012, 06:06 PM
Deveno
Re: Proving union and intersection of indexed family of sets
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?