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A set can be closed or not closed, not an individual element, such as zero.

Furthermore, closure depends on the operation under consideration.

Q: Why is zero not closed under the operation of whole numbers?

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Yes because being closed under an operation means that when the operation is performed on members of a set the result is also a member of the set, and when any two [members of the set of] whole numbers are added together the result of the addition is also a whole number which is, unsurprisingly, a member of the set of whole numbers.

The whole numbers are not closed under division! The statement is false since, for example, 2/3 is not a whole number.

No. Since -1 x -31 (= 31) would not be in the set.

Yes.

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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l think multiplication

The set of positive whole numbers is not closed under subtraction! In order for a set of numbers to be closed under some operation would mean that if you take any two elements of that set and use the operation the resulting "answer" would also be in the original set.26 is a positive whole number.40 is a positive whole number.However 26-40 = -14 which is clearly not a positive whole number. So positive whole numbers are not closed under subtraction.

Yes because being closed under an operation means that when the operation is performed on members of a set the result is also a member of the set, and when any two [members of the set of] whole numbers are added together the result of the addition is also a whole number which is, unsurprisingly, a member of the set of whole numbers.

The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.

The whole numbers are not closed under division! The statement is false since, for example, 2/3 is not a whole number.

A set of numbers is considered to be closed if and only if you take any 2 numbers and perform an operation on them, the answer will belong to the same set as the original numbers, than the set is closed under that operation. If you add any 2 real numbers, your answer will be a real number, so the real number set is closed under addition. If you divide any 2 whole numbers, your answer could be a repeating decimal, which is not a whole number, and is therefore not closed. As for 0 and 3, the most specific set they belong to is the whole numbers (0, 1, 2, 3...) If you add 0 and 3, your answer is 3, which is also a whole number. Therefore, yes 0 and 3 are closed under addition

No. Since -1 x -31 (= 31) would not be in the set.

The set of whole numbers is not closed under division (by non-zero whole numbers).

It depends on your definition of whole numbers. The classic definition of whole numbers is the set of counting numbers and zero. In this case, the set of whole numbers is not closed under subtraction, because 3-6 = -3, and -3 is not a member of this set. However, if you use whole numbers as the set of all integers, then whole numbers would be closed under subtraction.

If you can never, by multiplying two whole numbers, get anything but another whole number back as your answer, then, YES, the set of whole numbers must be closed under multiplication.

Yes.

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.