Which statement is true about the relationships between the number sets?(1 point)
Responses
Whole numbers include all natural numbers and 0.
Not all natural numbers are real numbers.
Some irrational numbers are also integers.
All rational numbers are also integers.
Whole numbers include all natural numbers and 0.
The statement that is true about the relationships between the number sets is:
Whole numbers include all natural numbers and 0.
To determine which statement is true about the relationships between the number sets, let's analyze each statement individually:
1. "Whole numbers include all natural numbers and 0."
To determine if this statement is true, we need to understand the definitions of whole numbers and natural numbers.
- Natural numbers include all positive integers (1, 2, 3, ... ) excluding 0 because they represent counting numbers.
- Whole numbers include 0 and all positive integers (0, 1, 2, 3, ... ).
Since whole numbers encompass all natural numbers (including 0), the statement "Whole numbers include all natural numbers and 0" is true.
2. "Not all natural numbers are real numbers."
To evaluate this statement, let's define real numbers and natural numbers.
- Natural numbers: These are positive integers representing counting numbers (1, 2, 3, ... ).
- Real numbers: These include both rational numbers (numbers that can be expressed as fractions) and irrational numbers (numbers that cannot be expressed as fractions).
Since natural numbers are a subset of real numbers (since they are counted among the positive integers), the statement "Not all natural numbers are real numbers" is false.
3. "Some irrational numbers are also integers."
Recall that irrational numbers cannot be expressed as fractions. On the other hand, integers consist of both positive and negative whole numbers along with zero (-∞, ... -3, -2, -1, 0, 1, 2, 3, ... +∞).
Since integers do encompass some irrational numbers (such as the square root of 2 or π), the statement "Some irrational numbers are also integers" is true.
4. "All rational numbers are also integers."
Rational numbers consist of both integers and fractions (a/b, where a and b are integers and b is not zero). Integers are a subset of rational numbers, but rational numbers also include fractions.
Thus, since not all rational numbers fall exclusively within the set of integers, the statement "All rational numbers are also integers" is false.
Therefore, the true statement about the relationships between the number sets is: "Whole numbers include all natural numbers and 0."