Assuming that you mean opposites (rather than oppisites), the answer will depend on what kind of opposites: additive opposites or multiplicative opposites.
In the first case you have the same set as you started with: whole numbers.
In the second, you will have the set that consists of the union of
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The set of all whole numbers and their opposites are
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.
The set of whole numbers includes all their additive opposites. So the set is the same as the one you started with.
It would be an integer (positive or negative whole number).
These are the integers.