Yes. When you add any whole numbers you get another whole number. That is what closed means in this context. The answer is still a whole number.
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Yes. The empty set is closed under the two operations.
Quite simply, they are closed under addition. No "when".
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
Yes.