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.
Note that N is an element of P(N), but cannot be an element of any P(Nn) (since these sets are all finite). Hence N is in every An, 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 An is indeed infinite.
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 .
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.