Basic sets

I'm having problems with questions g) and k).

For g), apparantly the answer is i) but it seems to me they're all the same.If |b'| is a finite subset of an infinite set, then |b| must be an infinite set as N is infinite. Is there any difference between saying "b is infinite" and "|b| = infinity"?

For k), apparantly the answer is i) as well, but I still don't understand the notation. Could someone explain it through?.

Page in question: http://i38.tinypic.com/142ev41.jpg

Ah, I get how k) works now, thanks.

But for g), how can you say ii) and iii) constitute the set of odd numbers? Couldn't they be even numbers? Or any infinite random subset? Now I think about it, ii) and iii) could be different, by being different subsets of N.