Yes.
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Get a proper subset of whole numbers.
A set can be closed or not closed, not an individual element, such as zero. Furthermore, closure depends on the operation under consideration.
Actually the set of integers is the same as the set of whole numbers since the whole numbers include negative whole numbers and zero.
Whole numbers and natural numbers are the exact same, except that whole numbers include zero
Apart from zero (which is its own opposite), the opposites of whole numbers are also whole numbers. You have the set of whole numbers which is also known as the set of integers.