Yes.
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The set of natural numbers is a subset of the set of whole numbers. The set of whole numbers is a subset of the set of integers. So the set of integers is the largest of these three sets.
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.
Integers are all positive and negative whole numbers, and natural numbers are all positve whole numbers including zero. So, natural numbers is a subset of integers.
You have it backwards. Integers are a subset of real numbers.
The Natural numbers is the set of Integers greater than 0 (ie {1, 2, 3, ...})