Let's evaluate each statement one by one:
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Whole numbers include all natural numbers and 0.
- This statement is true. Whole numbers are defined as the set of natural numbers (0, 1, 2, 3, ...) including zero.
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Some irrational numbers are also integers.
- This statement is false. By definition, integers are whole numbers that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...). Irrational numbers cannot be expressed as fractions or ratios of integers, so they cannot be integers.
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Not all natural numbers are real numbers.
- This statement is false. All natural numbers are considered real numbers. The set of real numbers includes all the rational numbers (which contain the integers and natural numbers) and all the irrational numbers.
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All rational numbers are also integers.
- This statement is false. Rational numbers are numbers that can be expressed as a fraction (p/q where p and q are integers and q ≠ 0). While all integers can be expressed as rational numbers (e.g., 3 can be expressed as 3/1), not all rational numbers are integers (e.g., 1/2 is a rational number but not an integer).
Based on this evaluation, the only true statement regarding the relationships between the number sets is:
- Whole numbers include all natural numbers and 0.