Choose all true statements.

All real numbers are rational numbers.

Some rational numbers are natural numbers.

No real numbers are irrational numbers.

All whole numbers are integers.

Some integers are natural numbers.

No rational numbers are integers.

recall that

natural ⊂ whole ⊂ integer ⊂ rational ⊂ real

Some rational numbers are natural numbers.

All whole numbers are integers.

Some integers are natural numbers.

To determine which statements are true, let's go through each one:

1. All real numbers are rational numbers.
This statement is false. Real numbers include both rational and irrational numbers. Rational numbers can be expressed as a fraction (e.g., 1/2, 3/4), while irrational numbers cannot be expressed as a fraction (e.g., square root of 2, pi). Therefore, not all real numbers are rational.

2. Some rational numbers are natural numbers.
This statement is true. Natural numbers are positive whole numbers (1, 2, 3, ...), and since rational numbers can include whole numbers, some rational numbers can also be natural numbers (e.g., 1/1, 2/1).

3. No real numbers are irrational numbers.
This statement is false. Irrational numbers are a subset of real numbers and cannot be expressed as fractions or ratios. Examples of irrational numbers include the square root of 2, pi, and Euler's number (e).

4. All whole numbers are integers.
This statement is true. Whole numbers include all positive integers (natural numbers) as well as zero (0). Integers encompass both positive and negative whole numbers (-3, -2, -1, 0, 1, 2, 3), so all whole numbers are indeed integers.

5. Some integers are natural numbers.
This statement is true. Natural numbers are a subset of integers, consisting of positive whole numbers (1, 2, 3, ...). Therefore, some integers can also be natural numbers.

6. No rational numbers are integers.
This statement is false. Rational numbers include any number that can be expressed as a fraction, where the numerator and denominator are both integers. Since integers fit this criteria (e.g., 1/1, 2/1), some rational numbers are indeed integers.

So, the true statements are:

- Some rational numbers are natural numbers.
- All whole numbers are integers.
- Some integers are natural numbers.