The hierarchy of number sets within the real number system from largest to smallest is best represented by option A. However, there is an inconsistency because whole numbers should be under integers, so let's break down the options provided:
- Natural numbers: {1, 2, 3, ...}
- Whole numbers: {0, 1, 2, 3, ...}
- Integers: {..., -2, -1, 0, 1, 2, ...}
- Rational numbers: numbers that can be expressed as a fraction (includes integers, whole numbers, and natural numbers)
- Irrational numbers: numbers that cannot be expressed as a simple fraction, including square roots of non-perfect squares and non-repeating, non-terminating decimals.
Considering this breakdown:
- Option A lists integers, rational numbers, natural numbers, and whole numbers. However, whole numbers are a subset of integers, which is incorrect.
- Option B is not a hierarchy of number sets; it's a list of types of irrational numbers.
- Option C has the natural numbers listed before whole numbers, which is incorrect.
- Option D lists rational numbers first as the largest set, but it incorrectly orders the subsets.
Given these observations, none of the options accurately provide a hierarchy. However, if we were to create an accurate hierarchy, it would look like:
- Real Numbers (includes both rational and irrational numbers)
- Rational Numbers (which includes integers, whole numbers, and natural numbers)
- Integers
- Whole Numbers
- Natural Numbers
- Irrational Numbers
So no correct option is present among the ones you listed, but based on the framework, I would conclude that they each have inaccuracies. If forced to choose one that somewhat aligns, A touches on the idea of hierarchy but is ultimately incorrect due to the errors in set placement.
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