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1. Which statement is true about the relationships between the number sets?

A. All irrational numbers are also integers.
B. Every number is a whole number.
C. Only positive numbers make up the set of integers.
D. Whole number include all natural numbers and 0.

3 answers

The true statement about the relationships between the number sets is:

D. Whole numbers include all natural numbers and 0.

To clarify the other options:

A. This is false because irrational numbers are not integers (e.g., √2, π). B. This is false because not every number is a whole number; for example, negative numbers and fractions are not whole numbers. C. This is false because integers include both positive and negative whole numbers, as well as zero (e.g., -1, 0, 1).

explain why

Certainly! Let’s break down the reasoning for each statement to understand why D is true and the others are false:

A. All irrational numbers are also integers.

Explanation:

  • Irrational Numbers: These are numbers that cannot be expressed as a simple fraction of two integers. Examples include √2, π, and e. They have non-repeating, non-terminating decimal expansions.
  • Integers: These are whole numbers that can be positive, negative, or zero (e.g., -3, 0, 5).
  • Conclusion: Since irrational numbers cannot be expressed as integers, this statement is false.

B. Every number is a whole number.

Explanation:

  • Whole Numbers: These are non-negative integers, which include 0 and all positive integers (0, 1, 2, 3, ...).
  • Other Types of Numbers: There are many types of numbers that are not whole numbers, such as negative integers (-1, -2, ...), fractions (1/2, 3/4), and irrational numbers (√2, π).
  • Conclusion: Since not every number is included in the set of whole numbers, this statement is false.

C. Only positive numbers make up the set of integers.

Explanation:

  • Integers: As previously mentioned, integers include positive numbers (1, 2, 3, ...), negative numbers (-1, -2, -3, ...), and zero (0).
  • Conclusion: Since integers also consist of negative numbers and zero, this statement is false.

D. Whole numbers include all natural numbers and 0.

Explanation:

  • Natural Numbers: These are typically the set of positive integers (1, 2, 3, ...).
  • Whole Numbers: This set includes all natural numbers plus zero (0, 1, 2, 3, ...).
  • Conclusion: The definition of whole numbers correctly incorporates all natural numbers along with zero, making this statement true.

In summary, option D accurately reflects the relationship between whole numbers and natural numbers, while options A, B, and C contain inaccuracies about the definitions of number sets.