The set of positive numbers, the set of negative numbers are two examples. Any subsets of these will also not be closed.
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Yes, the set is closed.
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
Yes. The empty set is closed under the two operations.
It is not closed under taking square (or other even) roots.
If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.