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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.

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Q: Which sets of numbers are closed under subtraction?

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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.

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.

Subtraction and addition are not properties of numbers themselves: they are operators that can be defined on sets of numbers.

Different sets of numbers have different properties. For example,The set of counting numbers is closed under addition but not under subtraction.The set of integers is closed under addition, subtraction and multiplication but not under division.Rational numbers are closed under all four basic operations of arithmetic, but not for square roots.A set S is "closed" with respect to operation # if whenever x and y are any two elements of S, then x#y is also in S. y = 0 is excluded for division.So, the answer depends on what you mean by "number".

I know that whole numbers, integers, negative numbers, positive numbers, and even numbers are. Anyone feel free to correct me.

Yes.

Yes

You can have counting number in multiplication and addition. All integers are in multiplication, addition and subtraction. All rational numbers are in all four. Real numbers, complex numbers and other larger sets are consistent with the four operations.

There are infinitely many sets of this type. Some of the common sets include natural numbers, integers, rational numbers, real numbers, complex numbers. Also, as an example, all sets of multiples of some whole number, for instance: { ... -6, -4, -2, 0, 2, 4, 6, ...} {... -9, -6, -3, 0, 3, 6, 9, ...} etc.

When you combine any two numbers in a set the result is also in that set. e.g. The set of whole numbers is closed with respect to addition, subtraction and multiplication. i.e. when you add, subtract or multiply two numbers the answer will always be a whole number. But the set of whole numbers is NOT closed with respect to division as the answer is not always a whole number e.g. 7÷5=1.4 The answer is not a whole number.

For example:* The set of real numbers, excluding zero * The set of rational numbers, excluding zero * The set of complex numbers, excluding zero You can also come up with other sets, for example: * The set {1} * The set of all powers of 2, with an integer exponent, so {... 1/8, 1/4, 1/2, 1, 2, 4, 8, 16, ...}

;: Th. Closed under union, concatenation, and Kleene closure. ;: Th. Closed under complementation: If L is regular, then is regular. ;: Th. Intersection: .

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