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Irrational Numbers, Rational Numbers, Integers, Whole numbers, Natural numbers
Starting at the top, we have the real numbers. The rational numbers is a subset of the reals. So are the irrational numbers. Now some rationals are integers so that is a subset of the rationals. Then a subset of the integers is the whole numbers. The natural numbers is a subset of those.
The natural numbers (ℕ) are a subset of the integers (ℤ) which are a subset of the rational numbers (ℚ) which are a subset of the real numbers (ℝ): ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ → ℕ ⊂ ℝ and ℤ ⊂ ℝ as well as ℚ ⊂ ℝ
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.