1. ## Families of Sets

I can't seem to manage to understand what these problems are asking for. I have to do 8 and 9, and am having a lot of trouble seeing what I am supposed to do. I've been looking at 8 a and 8c, but I still am not understanding how to do the rest. Any help would be greatly appreciated. I think I understand the concept of a family of sets, but I can't seem to get started on either of these problems.

2. ## Re: Families of Sets

For 8 have you tried just writing out a few of the sets to see what sets you are dealing with? For example, (b) says the sets are $\displaystyle \{x\in R: -\frac{1}{n}< x< 1\}, n\in N$.
For n= 1, that is $\displaystyle \{ x\in R: -1< x< 1\}$. For n= 2, $\displaystyle \{-\frac{1}{2}< x< 1\}$. For n= 3, $\displaystyle \{-\frac{1}{3}< x< 1\}$. Each of those sets is contained in the previous one. So the union is just the biggest set, the first one. The intersection will be numbers that are in all them. Of course, $\displaystyle -\frac{1}{n}$ goes to 0 as n goes to infinity. The right boundary is always 1.

For 9, $\displaystyle E_x$ is the set of all rational numbers between 0 and x. What happens as x gets larger and larger?

3. ## Re: Families of Sets

Originally Posted by renolovexoxo
I can't seem to manage to understand what these problems are asking for. I have to do 8 and 9, and am having a lot of trouble seeing what I am supposed to do. I've been looking at 8 a and 8c, but I still am not understanding how to do the rest. Any help would be greatly appreciated. I think I understand the concept of a family of sets, but I can't seem to get started on either of these problems.
The answers to 8a are: $\displaystyle \bigcup {{A_n} = \mathbb{R}} \;\& \;\bigcap {{A_n}} = ( - 1,1)$

The answers to 8c are: $\displaystyle \bigcup {{A_n} = (-1,2)} \;\& \;\bigcap {{A_n}} = [0,1]$

For #9 you need to realize that in $\displaystyle [0,x),~x\in(0,1)$ there is always a rational number.