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Let's analyze the given problem step by step.

The statement \( S \) is: "All natural numbers (positive integers) are real numbers."

1. **Original Statement \( S \)**: 
   - This is \( \text{true} \) because every natural number is indeed a real number.

2. **Converse of \( S \)**: 
   - The converse of the statement \( S \) is "All real numbers are natural numbers."
   - This is \( \text{false} \) because not all real numbers are natural numbers (e.g., fractions, negative numbers, irrational numbers).

3. **Contrapositive of \( S \)**:
   - The contrapositive of the statement \( S \) is "If a number is not a real number, then it is not a natural number."
   - This is \( \text{true} \) because if something is not a real number, it cannot be a natural number.

4. **Negation of \( S \)**:
   - The negation of the statement \( S \) is "Some natural numbers are not real numbers."
   - This is \( \text{false} \) because all natural numbers are real numbers.

Now, let's match these with the statements A-G:

- A. Numbers aren't natural.
- B. All real numbers are natural.
- C. Some natural numbers aren't real numbers.
- D. Some real numbers aren't natural numbers.
- E. No real number is natural.
- F. A number can’t be natural if it isn’t real.
- G. A number can’t be real if it isn’t natural.

Given the definitions:

- \( S \) (true or false): \( S \) is true.
- The converse of \( S \) (enter a letter from A-G and that statement is true or false): B, false.
- The contrapositive of \( S \) (enter a letter from A-G and that statement is true or false): F, true.
- The negation of \( S \) (enter a letter from A-G and that statement is true or false): C, false.

Let's fill in the blanks:

- \( S \) is (true or false): **true**.
- The converse of \( S \) is \( B \) and that statement is (true or false): **false**.
- The contrapositive of \( S \) is \( F \) and that statement is (true or false): **true**.
- The negation of \( S \) is \( C \) and that statement is (true or false): **false**.

Would you like more details on any part of this solution or have any questions?

Here are five relative questions to further explore the topic:
1. What are some examples of numbers that are real but not natural?
2. How would you define the inverse of a conditional statement?
3. Can a statement and its converse both be true? Provide an example.
4. Why is the contrapositive always logically equivalent to the original statement?
5. How do you negate a conditional statement correctly?

**Tip:** When dealing with logical statements, always remember that the truth of the original statement does not necessarily imply the truth of its converse.

Consider the statement 'A implies B'. The converse of this statement is 'B implies A'; its contrapositive is 'not B implies not A', and its negation is 'A and not B'. For example, consider the statement: all natural numbers (positive integers) are real numbers. This is a true statement, but this is not relevant for this discussion. The statement can be put as an implication: if z is a natural number, then z is a real number. The converse of this statement is all real numbers are natural numbers, or if z is a real number then it is a natural number (false). The contrapositive is if z is not a real number then it isn't a natural number (true). The negation is some natural numbers are not real numbers (false). Complete the sentences below, filling in A-G and T or F.

Explore the logical relationships between statements, their converses, contrapositives, and negations. This detailed guide will help students understand how to analyze mathematical statements and their truth values with clear examples and explanations, suitable for high school to early college levels.

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