a) 0 cannot be part of S because if it were then all three of , , and are true. But only one of them is allowed to be true.

b) I will simply note that the positive integers are closed under addition and multiplication. You can supply the rest of the details.

c) I would like to say the same thing about c) as I did about b): the positive rationals are closed under addition and multiplication. However, this set excludes the negative rationals, and I'm not sure exactly how to go about doing that. The exclusion of 0 from the set is going to be important, I can tell you that much.

-Dan