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Yes.

The natural numbers {1, 2, 3, ...} are all contained within the integers {..., -3, -2, -1, 0, 1, 2, 3, ...}.

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Counting numbers

Q: Are the natural numbers a subset of the integers?

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Yes. For example, the set of odd natural numbers is a infinite subset of the set of integers.

Mixed numbers and integers are different things.

All natural numbers are integers, not all integers are natural numbers.

Traditionally, the set of integers that represents the natural numbers is {1,2,3,...}, which are the positive integers. Some people include the non-negative integers as the set of natural numbers, which is {0,1,2,3,...}, and includes 0.

The set of positive integers contains 1 but not zero. Within the set of integers, there is the subset of positive integers, the subset of negative integers and the subset with a single element in it - zero. There are a zillion other sets that could be specified that meet the conditions set down in the question. The one cited is an easy one.

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Natural numbers are a subset of the set of integers, among others.

Yes.

Positive integers are (not is!) a proper subset of natural numbers. The natural numbers comprise positive integers and zero.

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.

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.

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 a subset of rational numbers which are a subset of real numbers which are a subset of complex numbers ...

No, integers are a subset of rational numbers.

Natural numbers = Whole numbers are a subset of integers (not intrgers!) which are a subset of rational numbers. Rational numbers and irrational number, together, comprise real numbers.

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, ...})

No, because there are no naturl numbers.But natural numbers (even), are not equivalent to integers: they are a subset of integers. Negative whole numbers are integers but they are not natural numbers. Mathematicians are undecided about whether or not zero is a natural number.