Verified Answer :

  • Natural Numbers (\(N\)): \(52, 8, 1\)
  • Integers (\(Z\)): \(-12\)
  • Rational Numbers (\(Q\)): \(3 \frac{1}{2}, -2.4, \frac{5}{8}, -4.92\)
  • Real Numbers (\(R\)): All numbers listed.

Explanation

  1. Definitions and Hierarchy of Number Sets

To categorize the set of numbers \(\{52, 3 \frac{1}{2}, -12, 8, -2.4, \frac{5}{8}, 1, -4.92\}\), we must first identify the definitions of the standard number sets:

  • Natural Numbers (\(N\)): These are the positive counting numbers, \(\{1, 2, 3, \dots\}\).
  • Integers (\(Z\)): These include all natural numbers, their negatives, and zero, \(\{\dots, -2, -1, 0, 1, 2, \dots\}\).
  • Rational Numbers (\(Q\)): These are numbers that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q eq 0\). This category includes all integers, terminating decimals, and fractions.
  • Real Numbers (\(R\)): This encompasses all numbers in our list, as all provided values are real.

The relationship between these sets is hierarchical: \(N \subset Z \subset Q \subset R\). Our objective is to place each number into its most specific category.

  1. Categorizing the Numbers

We analyze each number to determine the most specific category to which it belongs:

  • \(52\): This is a positive integer, so it is a Natural Number (\(N\)).
  • \(3 \frac{1}{2}\): This is a fraction \(\frac{7}{2}\). It is not an integer, but it is a Rational Number (\(Q\)).
  • \(-12\): This is a negative integer, so it is an Integer (\(Z\)).
  • \(8\): This is a positive integer, so it is a Natural Number (\(N\)).
  • \(-2.4\): This is a terminating decimal \(\frac{-24}{10} = \frac{-12}{5}\). It is a Rational Number (\(Q\)).
  • \(\frac{5}{8}\): This is a fraction. It is a Rational Number (\(Q\)).
  • \(1\): This is a positive integer, so it is a Natural Number (\(N\)).
  • \(-4.92\): This is a terminating decimal \(\frac{-492}{100} = \frac{-123}{25}\). It is a Rational Number (\(Q\)).
  1. Conclusion and Summary of Categorization

Based on the analysis, we group the numbers into their most specific sets:

  • Natural Numbers (\(N\)): \(52, 8, 1\)
  • Integers (\(Z\)): \(-12\)
  • Rational Numbers (\(Q\)): \(3 \frac{1}{2}, -2.4, \frac{5}{8}, -4.92\)

Note: All listed numbers are also Real Numbers (\(R\)). We have successfully classified all items in the given set.

Examples

Understanding number sets is fundamental in finance and data science. For instance, if a bank system tracks account balances, it might store integers for account IDs (as they are discrete and positive) and rational numbers (or decimals) for the balance amounts (as they represent fractions of currency). Classifying data types accurately ensures that calculations like interest rates or tax deductions are performed with the correct precision, preventing errors in financial reporting.

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