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Yes - the set of integers is a subset of the set of rational numbers.
The set of integers is a proper subset of the set of rational numbers.
Counting numbers are a proper subset of whole numbers which are the same as integers which are a proper subset of rational numbers.
Whole numbers are the same as integers. Whole numbers are a proper subset of rational numbers.
Whole numbers and integers are the same thing. They are a proper subset of rational numbers.