A set is closed under a particular operation (like division, addition, subtraction, etc) if whenever two elements of the set are combined by the operation, the answer is always an element of the original set. Examples: I) The positive integers are closed under addition, because adding any two positive integers gives another positive integer. II) The integers are notclosed under division, because it is not true that an integer divided by an integer is an integer (as in the case of 1 divided by 5, for example). In this case, the answer depends on the definition of "whole numbers". If this term is taken to mean positive whole numbers (1, 2, 3, ...), then the answer is no, they are not closed under subtraction, because it is possible to subtract two positive whole numbers and get an answer that is not a positive whole number (as in the case of 1 - 10 = -9, which is not a positive whole number)
Yes, the set of integers is closed under subtraction.
True
A set of real numbers is closed under subtraction when you take two real numbers and subtract , the answer is always a real number .
Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.
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.
The set of rational numbers is closed under all 4 basic operations.
Please clarify what set you are talking about. There are several sets of numbers. Also, "closed under..." should be followed by an operation; "natural" is not an operation.
No.
Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
A set of real numbers is closed under subtraction when you take two real numbers and subtract , the answer is always a real number .
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.
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.
Yes, they are.
Real numbers are closed under addition and subtraction. To get a number outside the real number system you would have to use square root.
Subtraction.
Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.
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.
The set of rational numbers is closed under all 4 basic operations.
No; here's a counterexample to show that the set of irrational numbers is NOT closed under subtraction: pi - pi = 0. pi is an irrational number. If you subtract it from itself, you get zero, which is a rational number. Closure would require that the difference(answer) be an irrational number as well, which it isn't. Therefore the set of irrational numbers is NOT closed under subtraction.