What Are the Intersections and Unions in Indexed Families of Sets?

[merlan]

Let $$B_n = (0, \frac {1}{n} ]$$ for all $$n \in N$$ (N = set of natural numbers)

a) For each $$n \in N$$, find $$\bigcap _{k=1}^n B_k$$ and $$\bigcup _{k=1}^n B_k$$

b) Find $$\bigcap _{n=1}^ \infty B_n$$ and $$\bigcup _{n=1}^ \infty B_n$$

For a) I have
$$B_1 = (0,1] \\ B_2 = (0, \frac {1}{2} ] \\ B_3 = (0, \frac {1}{3} ]$$

so $$\bigcap _{k=1}^n B_k$$ appears to be $${ \emptyset }$$ and $$\bigcup _{k=1}^n B_k$$ looks like $$(0,1]$$

I'm new to this and any help would be greatly appreciated. The questions I have are is a) correct? and what is the difference between a) and b)?

 

[AKG]

Question a) asks you for the intersection/union of n sets, question b) asks you for the intersection/union of an infinite number of sets. How are you getting the answers you're getting?

 

[merlan]

I started with n=1 which produced the interval (0,1], then I did n=2 and then n=3. For the union I took the smallest x and the largest y, which yielded (0,1] and for the intersection I took the largest x and the smallest y and saw that we would never reach the smallest so no intersection.

I see now that this is wrong for a) but is it the correct process for b)?

 

[AKG]

Nothing you wrote makes sense.

I started with n=1 which produced the interval (0,1]

What exactly "produced" the interval (0,1]?

then I did n=2 and then n=3

What did you do with n=2, n=3?

For the union I took the smallest x and the largest y, which yielded (0,1]

What are x and y? What exactly did you do that yielded (0,1]?

and for the intersection I took the largest x and the smallest y and saw that we would never reach the smallest so no intersection.

This just makes no sense. Again, what are x and y? What do you mean, "we could never reach the smallest"? How does this mean that there is no intersection?

I see now that this is wrong for a) but is it the correct process for b)?

What process? I have no idea what you're doing. Do you? At all? Are there examples in your textbook? Are you sure you even know what the symbols you're working with mean?

$$\bigcup _{k=1} ^n B_k$$

is the union of B₁, B₂, ..., B_n. It consists of all the elements that are in at least one of the above listed sets.

$$\bigcap _{k=1} ^n B_k$$

is the intersection of B₁, B₂, ..., B_n. It consists of all the elements that are in each and every one of those above listed sets. What numbers are in each of:

(0,1], (0,1/2], ..., (0,1/n]?

For example, if n=5, which numbers are in each and every one of:

(0,1], (0,1/2], (0,1/3], (0,1/4], (0,1/5]?

As it stands, it appears that you're claiming that no number is in all of those sets. 1/6 is in all of them. So is 0.12344523. So is 0.1999. What about 0.2001?

 

[merlan]

I'm sorry if I have made you upset. I'm positive I am not explaining my thoughts correctly and I am also positive that I am confused on this topic, which is why I'm seeking help.

My $$B_1$$ is the same as yours, as is my $$B_2$$. I used the same process you did to get those. If my understading is correct $$(0,1]$$ for $$B_1$$ is an interval and I was trying to represent the start of the interval (x) and the end of the interval (y). It appears that I was trying to answering (b) instead of (a).

I think I see now how the two questions are different, atleast when you pick an n. I quess I was thinking that because there are infinitely many $$N$$ so the questions were the same.

 

[AKG]

Yes, you're definitely confused. Again, not much of what you said made sense. That fact that you say, "My B₁ is the same as yours, as is my B₂," suggests to me that you're incredibly lost. Let's just start with an incredibly simple example:

Let x = 4 and y = 5, find xy. Now it certainly wouldn't be meaningful to say, "okay, my x is the same as yours, as is my y." x and y are given, they are part of the problem statement. Now if I said that my xy was 21 and you said yours was too, then that would be meaningful. We'd both be wrong, but hopefully you see the difference.

In your problem, the idea is the same. For each n, B_n is defined as (0,1/n]. It's not up to us to figure out what B₁ and B₂ are, they're given. We want to find what the intersection of the first n of them are, for example. So we are on to our next example:

Instead of defining some sets for every n, let's just do it for 1 and 2. So, family of sets will not be indexed by N, i.e. {1, 2, 3, ...} but simply by {1, 2}. Let B₁ = {0, 1} and B₂ = {0}. Now:

$$\bigcup _{k=1} ^2 B_k = B_1 \cup B_2 = \{0,1\} \cup \{0\} = \{0,1\}$$

$$\bigcap _{k=1} ^2 B_k = B_1 \cap B_2 = \{0,1\} \cap \{0\} = \{0\}$$

Do you understand that much?

Now, do you know what (0,1/n] is? It is the set of real numbers greater than 0 and less than OR equal to 1/n.

Don't even think about question b), just focus on question a). Can you figure out the following:

$$(0,1] \cup (0,1/2] \cup (0,1/3]$$

$$(0,1] \cap (0,1/2] \cap (0,1/3]$$

Give me your answers to these questions, and you should probably show your work. Before you do it, heavily review whatever you have learned in your course so far. You should be very comfortable with finding the intersection of two sets, and should be very comfortable with dealing with interals like (x,y]. I also have to ask, why was this posted in calculus and beyond? Have you actually done calculus yet?

 

[merlan]

Yes, I've completed calculus, differential equations, linear algebra, a stat class and so on. I understand unions, intersections, intervals, functions, what natural numbers are, what sets are, however I was confused by indexed families of sets. This is why I came here, if I wasn't confused on this topic I would not have came here. At any rate I thank you for your time and in spite of your comments I see where I have been getting confused.
P.S.
Though your last comment about posting in calculus and beyond was a clever jab is it not true that the subject decides the section not what classes a poster has taken?

 

[AKG]

Though your last comment about posting in calculus and beyond was a clever jab is it not true that the subject decides the section not what classes a poster has taken?

This was not meant to be a clever jab, I'm sorry if I had offended you. You had been unclear and that frustrated me, so I may have been less courteous than I should have been. Yes, the subject decides the section, but the subject of this post does not appear to be calculus or beyond.