Indexed Collection of Sets ((

[rocomath]

Indexed Collection of Sets ... :(((

My notes are confusing me so bad, worst part is that we're not using our book till later on and that just kills me a lot. I'm very text-book because notes never make sense to me.

Let $$A_n=[0,n]$$

a) What numbers are in $$\bigcup_{n=1}^{\infty}A_n$$?

b) What numbers are in $$\bigcap_{n=1}^{\infty}A_n$$?

Ok, so I have an example here ...

If

$$I=\{1,2,3...\}$$

$$A_i=[-i,i]$$

Then

$$\bigcup=\mathbb{R}$$

$$\bigcap=[-1,1]$$

I honestly, can't remember how we got [-1,1]?

a) Since An goes from 0 to n, wouldn't that make $$\bigcup=[0,\infty)$$?

 

[Dick]

Sure. The union is [0,infinity). What's the intersection? In your example, the intersection is [-1,1] because the smallest index is 1. I'm not sure what's confusing you so seriously here? Draw a picture of the sets.

 

[rocomath]

Wouldn't my intersection for my example also be all real numbers? I could choose the smallest index within my set, which is 1. But I could continue choosing within the real number system, giving me an infinite amount of options.

$$\bigcap=\mathbb{R}$$?

 

[Dick]

Wouldn't my intersection for my example also be all real numbers? I could choose the smallest index within my set, which is 1. But I could continue choosing within the real number system, giving me an infinite amount of options.

$$\bigcap=\mathbb{R}$$?

No, no, no. The smallest set in your system is A_1 All of the others include it. An element of the intersection has to be in ALL of the sets. The intersection is [0,1].

 

[rocomath]

OHHH! In ALL the sets, makes a lot of sense now.

Because once I move onto n=2, I would have 0, 1, 2, but going back to n=1, it doesn't contain 2.

YAYYY :) Thanks!