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

>>22
And it's Korner who decided that his first years didn't have to do examples. I wasn't saying anything about the man himself. I wouldn't pretend to know him, but I've read some of the material on his homepage and he writes very lucidly and intelligently, which endears me to him.