Power Set of N-Intersection of infinitely many finite subsets of N

Hello everyone,

This is a problem I'm having a hard time solving:

Let = the power set of and for every natural number, n, be the power set of which is a **finite **subset of .

Now, if for every n, ,

what can be said about

Is that an infinite subset of ?

Thanks, in advance.

Re: Power Set of N-Intersection of infinitely many finite subsets of N

Quote:

Originally Posted by

**Neutriiino** This is a problem I'm having a hard time solving:

Let

= the power set of

and for every natural number, n,

be the power set of

which is a

**finite **subset of

.

Now, if for every n,

,

what can be said about

Is that an infinite subset of

?

We realize that this is your first posting. Nonetheless, it is best if you show some work so that we know where to begin.

If is a **finite subset** of is it possible for

If is an **infinite subset** of must for

Tell us what you think?

Re: Power Set of N-Intersection of finite subsets of N

Note that N is an element of P(N), but cannot be an element of any P(N_{n}) (since these sets are all finite). Hence N is in every A_{n}, and thus in the intersection.

The same exact logic hold for N - S, where S is any finite subset of N. Thus any co-finite subset of N is in the infinite intersection. Since we obtain a co-finite subset of N from any finite set S, simply by taking N - S, there are at least as many co-finite subsets of N, as there are finite subsets of N. Hence the intersection of all the A_{n} is indeed infinite.

Re: Power Set of N-Intersection of finite subsets of N

Thanks Deveno, I got it!

Thank you too, Plato, for your rather Socratic method of instruction through questions. :)

well, here we go:

If F is a finite subset of , then , where nmax is the largest number in F. hence , (for any value of n)

If G is an infinite subset of N, for every n, , hence .

Right?

I found this in a previously held exam in our department, and now I'm wondering where I can find more problems like this.

Thanks, once again.