The answer depends on what is meant by "their opposites". If you mean additive opposites then the set is of all non-zero integers.
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The set of numbers that include the natural numbers, their opposites and 0 is called the set of integers.
Whole numbers are the set of natural or counting numbers inclding zero
These are known as integers.
No. One, a counting number, doesn't belong to either of those sets.
This set of numbers is called "Whole Numbers".