If you interpret "whole numbers" as "integers", then yes. If you interpret "whole numbers" as "non-negative integers", then no.
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There is no counterexample because the set of whole numbers is closed under addition (and subtraction).
They form a closed set under addition, subtraction or multiplication.
Integers are closed under subtraction, meaning that any subtraction problem with integers has a solution in the set of integers.
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.
No, whole numbers are not closed under division. It is possible to divide one whole number by another whole number and get a result which is not a whole number, for example, 1/2. One divided by two is a half.