Which statement is true about the relationships between the number sets? (1 point)
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.
Whole numbers include all natural numbers and 0.
To determine which statement is true about the relationships between the number sets, let's examine each statement:
1. "Not all natural numbers are real numbers."
To figure out if this statement is true, we must understand the definitions of natural numbers and real numbers. Natural numbers are the set of positive integers starting from 1 (1, 2, 3, 4, ...). Real numbers, on the other hand, include all rational and irrational numbers. Since the set of natural numbers is a subset of the set of real numbers, this statement is true.
2. "Some irrational numbers are also integers."
To determine if this statement is true, we need to understand the definitions of irrational numbers and integers. Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers, such as √2 or π. Integers, on the other hand, include all whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...). Since irrational numbers and integers are distinct sets, there is no overlap, so this statement is false.
3. "All rational numbers are also integers."
To decide on the accuracy of this statement, we need to understand the definitions of rational numbers and integers. Rational numbers are numbers that can be expressed as a fraction or ratio of two integers, such as 1/2 or -3/4. Integers, as mentioned above, include all whole numbers and their negatives. Since all integers can indeed be expressed as a fraction with a denominator of 1 (e.g., 4 = 4/1), this statement is true.
4. "Whole numbers include all natural numbers and 0."
To verify the validity of this statement, let's consider the definitions of whole numbers and natural numbers. Whole numbers are the set of positive integers (1, 2, 3, ...) along with zero (0). Natural numbers, as discussed earlier, comprise the set of positive integers starting from 1 (1, 2, 3, ...). Since zero is not included in the set of natural numbers, this statement is true.
In summary, out of the four given statements, two are true:
- "Not all natural numbers are real numbers."
- "All rational numbers are also integers."
The correct statement is:
Whole numbers include all natural numbers and 0.