Q: Under what operation is the set of positive rational numbers not closed?

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To be closed under an operation, when that operation is applied to two member of a set then the result must also be a member of the set. Thus the sets ℂ (Complex numbers), ℝ (Real Numbers), ℚ (Rational Numbers) and ℤ (integers) are closed under subtraction. ℤ+ (the positive integers), ℤ- (the negative integers) and ℕ (the natural numbers) are not closed under subtraction as subtraction can lead to a result which is not a member of the set.

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.

No. It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.

No.

No. For a set to be closed with respect to an operation, the result of applying the operation to any elements of the set also must be in the set. The set of negative numbers is not closed under multiplication because, for example (-1)*(-2)=2. In that example, we multiplied two numbers that were in the set (negative numbers) and the product was not in the set (it is a positive number). On the other hand, the set of all negative numbers is closed under the operation of addition because the sum of any two negative numbers is a negatoive number.

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Yes, they are.

subtraction. Let's take 1/2 and subtract 3/4 which is great than 1/2 so the answer is negative and hence not a positive rational.

If a set is closed under an operation. then the answer will be a part of that set. If you add, subtract or multiply any two rational numbers you get another national number. But when it comes to division, it is closed except for one number and that is ZERO. eg 3.56 (rational number) ÷ 0 = no answer. Since no answer is not a rational number, that rational numbers are not closed under the operation of division.

2 = 2/1 is rational. Sqrt(2) is not rational.

To be closed under an operation, when that operation is applied to two member of a set then the result must also be a member of the set. Thus the sets ℂ (Complex numbers), ℝ (Real Numbers), ℚ (Rational Numbers) and ℤ (integers) are closed under subtraction. ℤ+ (the positive integers), ℤ- (the negative integers) and ℕ (the natural numbers) are not closed under subtraction as subtraction can lead to a result which is not a member of the set.

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.

No. It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.

Yes. In general, the set of rational numbers is closed under addition, subtraction, and multiplication; and the set of rational numbers without zero is closed under division.

Rational numbers are closed under multiplication, because if you multiply any rational number you will get a pattern. Rational numbers also have a pattern or terminatge, which is good to keep in mind.

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.

The set of integers, rational numbers, real numbers, complex numbers are some of the sets. Also, many of their subsets: for example, all numbers divisible by 3.

Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.