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Q: Are polynomial expressions closed under subtraction?

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Yes.

Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.

It means that you can do any of those operations, and again get a number from the set - in this case, a polynomial. Note that if you divide a polynomial by another polynomial, you will NOT always get a polynomial, so the set of polynomials is not closed under division.

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

Yes.

No.

Yes, the set of integers is closed under subtraction.

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.

A set of real numbers is closed under subtraction when you take two real numbers and subtract , the answer is always a real number .

Why, yes, it is.

Yes, it is.

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.

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.

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.

It means whatever members of the set you subtract, the answer will still be a member of the set. For example, the set of positive integers is not closed under subtraction, since 3 - 8 = -5

The set of rational numbers is closed under all 4 basic operations.

They are closed under all except that division by zero is not defined.

I Think is natural number a closed set under subtraction.

Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.

Integers are closed under subtraction, meaning that any subtraction problem with integers has a solution in the set of integers.

They form a closed set under addition, subtraction or multiplication.

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.