Yes, because for any x and y that are positive fractions (y not equal to zero), x/y is also a positive fraction. Note that whole numbers are considered fractions with denominators of 1 -- otherwise it doesn't work.
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No, the natural numbers are not closed under division. For example, 2 and 3 are natural numbers, but 2/3 is not.
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Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.
The set of rational numbers is closed under all 4 basic operations.
Division by 0, which can also be written as 0.000... (repeating) is not defined.