A subset of integers.
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No. Prime numbers are a subset of whole numbers.
A whole number is not a set of any kind and so cannot be a subset 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.
Natural numbers are just whole positive numbers. Since whole positive numbers can represent a distance along a line, they are a subset of real numbers.