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division
None.
Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.
The set of rational numbers is closed under all 4 basic operations.
You don't say that "an integer is closed". It is the SET of integers which is closed UNDER A SPECIFIC OPERATION. For example, the SET of integers is closed under the operations of addition and multiplication. That means that an addition of two members of the set (two integers in this case) will again give you a member of the set (an integer in this case).
division
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.
None.
Yes, because there is no way of multiplying two polynomials to get something that isn't a polynomial.
Yes, decidable languages are closed under operations such as union, intersection, concatenation, and complementation. This means that if a language is decidable, performing these operations on it will result in another decidable language.
Yes.
Yes. The empty set is closed under the two operations.
The set of rational numbers is closed under all 4 basic operations.
In the context of string operations, being closed under concatenation means that when you combine two strings together, the result is still a valid string. This property is important because it ensures that string operations can be performed without creating invalid or unexpected results.
In my opinion the question is poorly defined, since "non-polynomial" could be just about anything.
you dont
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.