whoch list shows an accurate hierarchy of number sets within the real number system

Answer 1

In the real number system, the hierarchy of number sets can be organized as follows, from the most specific to the most general:

1. Natural Numbers (ℕ): {1, 2, 3, ...} (sometimes including 0: {0, 1, 2, 3, ...})
2. Whole Numbers: {0, 1, 2, 3, ...} (sometimes used interchangeably with natural numbers depending on context)
3. Integers (ℤ): {..., -3, -2, -1, 0, 1, 2, 3, ...}
4. Rational Numbers (ℚ): Numbers that can be expressed as a fraction a/b where a and b are integers and b ≠ 0 (e.g., 1/2, -3, 0.75)
5. Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π, e)
6. Real Numbers (ℝ): All rational and irrational numbers combined

So, an accurate hierarchy of number sets within the real number system would be:

Natural Numbers ⊆ Whole Numbers ⊆ Integers ⊆ Rational Numbers ⊆ Real Numbers

(Note: The "Irrational Numbers" are not a separate set within this hierarchy; they are part of the Real Numbers and are a subset of the Rational Numbers' complement.)