I did not want to start 6 different threads so I am posting them all here since they all refer to the same problem which is:

Find the union and intersection of the collection of sets.

Any help on any of the problems would be great. I think I've gotten an idea for some of them. I'm stumped on 2 of the 6, and have some work for the rest.

(1)

(N is the set of natural numbers)

Sets have the form: (-1/n , 1) meaning all real numbers in that interval

= (-1, 1) because -1/n gets smaller as n gets larger so -1/n gets closer to 0. So the largest of sets will contain all other sets so that's how I got the union to be (-1, 1).

= [0, 1) because 0 will be in all sets and it will never reach 1 because of the definition of An

(2) (Q is the set of rational numbers)

Sets have the form [x, x + 1] meaning all rationals in that interval

= (-∞, ∞) because I can put any real number in for x and so the union would have to contain all of the sets and since the reals go on and on both ways I saw (-∞, ∞) as the union.

= because I couldn't find or "see" anything that all sets had in common

(3) (Q is the set of rational numbers)

Sets have the form: [r , r + 1] meaning all real numbers in that interval

-- this is one the questions that stumped me. I am having a hard time seeing the union and intersection. This collection of sets does look very similar to the previous problem but it switches the use of real and rational numbers. I tried to list out some sets (such as E0, E-1/2, E3/5, E3) to get some ideas, but I still couldn't come to some answer.

(4)

Sets have the form:

To get an idea, I wrote some sets out such as E.00001, E.00052, E.123, and E.9999.

So the end points of the intervals get closer and closer to -1 and 1 but never equal -1 and 1 because x is in (0, 1). So I got that:

= (-1,1) and = {0} since 0 is in all intervals of the form [(the negative of a number), (the number)].

(5) (N is the set of naturals)

Sets have the form: [0, 1/n) meaning all irrationals in that interval

The right side of the interval is getting smaller as n increases so [0, 1) is the largest which will contain all other smaller sets so I got:

= [0,1) and = {0} because 0 is in all the sets by the form [0, 1/n).

(6) (R\Q is the set of all irrational numbers)

This is the other question that stumped me. I could not figure out the form the sets would be in. Is the form "z < n"? And where is this n coming from?

I tried to list some sets but couldn't see a pattern nor get an idea towards an answer.

For example = so that means it is the set of all naturals greater than -√3 ?

I am quite confused with this one.

Thank you for your time.

Any help in any form (hints, tips, suggestions, corrections, etc.) is greatly appreciated!