No.
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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.
Real numbers are closed under addition and subtraction. To get a number outside the real number system you would have to use square root.
The answer depends on how the numbers are combined: addition, subtraction or multiplication.
Integers are closed under subtraction, meaning that any subtraction problem with integers has a solution in the set of integers.
No.