Rings: Don't quite understanding the basics...

Hi,

I can't quite get my head round rings, fields etc.

I can understand that for a ring, for example, the elements have to obey fundamental laws such as associate/commutative addition & multiplication, inverses etc, and this makes sense to me when I think of a set of integers modulo n, for example.

In a book that I'm reading it states ... that the underlying set of a ring need not be a set of numbers, and then it says we might take any set 'T' and let S = {all subsets of T}.

then define, for a & b which are members of S:

a + b = (a intersection b) - (a union b), and

a.b = a union b

So, I'm starting to see that the 'meaning' of '+' and '-' can be applied to a set ... I think? and I'm having trouble quite grasping what is going on here. It's like the fundamental laws are being applied to sets, and we can also define what the operations '+' and '-' mean when used in a Set context.

I'm just not quite getting the bigger picture here (not necessarily with regard to the example above, just in general). Can someone help me to understand please.

Many Thanks.

JG.