) If p→ (p→ q ) is false, then find the truth values of p and q.


To solve this, we need to analyze the given expression \( p \to (p \to q) \) and determine when it is false.

1. First, recall the truth table for the implication \( p \to q \):
   - \( p \to q \) is **false** only when \( p \) is true and \( q \) is false.
   - Otherwise, \( p \to q \) is **true**.

Now, we want \( p \to (p \to q) \) to be **false**. This means:
- \( p \) must be **true** (since \( p \to (p \to q) \) is false only when the antecedent is true and the consequent is false).
- \( (p \to q) \) must be **false** for the entire implication to be false.

For \( (p \to q) \) to be false:
- \( p \) must be **true** and \( q \) must be **false**.

Thus, the truth values are:
- \( p = \text{true} \)
- \( q = \text{false} \)

### Conclusion:
For \( p \to (p \to q) \) to be false, \( p \) must be **true** and \( q \) must be **false**.

Would you like to see the detailed truth table or explanation for the implication?

Here are 5 related questions:
1. What is the truth table for \( p \to q \)?
2. How does double implication (biconditional) differ from implication?
3. Can an implication be false when the consequent is true?
4. How would you express \( p \to (p \to q) \) using logical operators like AND and OR?
5. In which logical systems does implication behave differently?

**Tip:** When analyzing implications, focus on when the antecedent is true and the consequent is false—this is the only time the implication is false!

If p → (p → q) is false, then find the truth values of p and q.

Discover how to determine the truth values of p and q when the expression p → (p → q) is false. This problem explores the fundamentals of propositional logic and implications, suitable for students in Grades 9-12.

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