The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.
That is correct, the set is not closed.
Any time you add integers, the sum will be another integer.
They are closed under all except that division by zero is not defined.
Because the set is not closed under addition. If x and y are odd, then x + y is not odd.
If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.
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).
negetive integers are not closed under addition but positive integers are.
The set of integers is not closed under multiplication and so is not a field.
No, it is not.