Yes, that is part of the definition of a group.
A set can be closed or not closed, not an individual element, such as zero. Furthermore, closure depends on the operation under consideration.
Yes.
No, a set of natural numbers is not a group under the operation of addition. For a set to be a group, it must satisfy four properties: closure, associativity, identity, and inverses. While the natural numbers are closed under addition and associative, there is no additive identity (0 is not included in the natural numbers) and no inverses (there is no natural number that can be added to another natural number to yield zero).
No. It is not a group.
Division, since you can't divide by zero.
Yes, the Canadian Football League (CFL) is not closed under union operation.
A set can be closed or not closed, not an individual element, such as zero. Furthermore, closure depends on the operation under consideration.
No. It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.
Addition.
addition
yes
Yes, they are.
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.
Subtraction.
Yes.
l think multiplication
Yes, the set of non-deterministic polynomial time (NP) problems is closed under the operation of union.