I going to denote the operations in this manner:

Addition → A

Subtraction → S

Multiplication → M

Division → D

And let O denote any of the above operations.

(For example, aOb may represent a+b or a-b or a×b or a÷b (only when b≠0)).

For any set T, operation O is said to be closed in set T if for all a, b ∈T ⇒ aOb∈T.

Answers to the questions:

1. A, M, D

2. A, M, D

3. M

4. A, M, D

5. None

6. None

7. M, D

8. None

9. None

10. A, M

11. A, M

12. A, S, M

13. A, S, M

14. A, S, M, D

15.A, S, M, D

Additional question:

Rational number set (Q) : A, S, M, D

Irrational number set (Q') : A, S