Understanding the Hierarchical Chain of Number Systems

Number systems are organized in a specific hierarchical structure, meaning some sets of numbers are subsets of others. Let's define the number systems mentioned in the question:

• Natural Numbers (N): These are the counting numbers starting from 1. $N = \{1, 2, 3, 4, ...\}$.
• Whole Numbers (W): These include all natural numbers and zero. $W = \{0, 1, 2, 3, 4, ...\}$.
• Integers (Z): These include all whole numbers and their negative counterparts. $Z = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.
• Rational Numbers (Q): These are numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. Examples include $\frac{1}{2}$, $-3.5$ (which is $-\frac{7}{2}$), and $4$ (which is $\frac{4}{1}$).

Identifying the Relationship Between Number Sets

Now, let's look at how these sets relate to each other:

• Every natural number is also a whole number. For example, 3 is in N and 3 is also in W. So, Natural Numbers (N) are a subset of Whole Numbers (W). We can write this as $N \subset W$.
• Every whole number is also an integer. For example, 0 is in W and 0 is also in Z. Every natural number is also an integer. So, Whole Numbers (W) are a subset of Integers (Z). We can write this as $W \subset Z$.
• Every integer is also a rational number. For example, -5 is in Z and -5 can be written as $\frac{-5}{1}$, which is a rational number. So, Integers (Z) are a subset of Rational Numbers (Q). We can write this as $Z \subset Q$.

Constructing the Hierarchical Chain

Putting these relationships together, we see a clear chain:

Natural Numbers (N) are inside Whole Numbers (W).

Whole Numbers (W) are inside Integers (Z).

Integers (Z) are inside Rational Numbers (Q).

This forms the hierarchical chain:

$N \subset W \subset Z \subset Q$

This inclusion can be represented by arrows indicating "is a subset of" or "leads to the next larger set". The correct sequence representing this hierarchy is $N \rightarrow W \rightarrow Z \rightarrow Q$.

Evaluating the Given Sequences

Let's examine the provided options based on our understanding of the number system hierarchy:

Based on the analysis, the sequence N → W → Z → Q accurately represents the hierarchical chain where each set is a subset of the next one in the sequence.