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The image contains logical propositions involving the truth values of \( p \), \( q \), and \( r \). Here's a breakdown of the questions in the image:

1. **Problem 5**: You are asked to determine the truth values of several logical propositions given that:
   - \( p \) is true (\( V \)),
   - \( q \) is true (\( V \)),
   - \( r \) is false (\( F \)).

### Let's work through the propositions in **5.1 - 5.4**:

- **5.1**: \((p \Rightarrow q) \land (p \lor q)\)

   - \( p \Rightarrow q \) is true because both \( p \) and \( q \) are true (true implies true is true).
   - \( p \lor q \) is true because both \( p \) and \( q \) are true (OR between two truths is true).
   - Therefore, \( (p \Rightarrow q) \land (p \lor q) = V \land V = V \).

   **Answer**: Verdadero (True)

- **5.2**: \((\neg p \lor q) \Rightarrow (p \land r) \lor p\)

   - \( \neg p \) is false since \( p \) is true, so \( \neg p \lor q = F \lor V = V \).
   - \( (p \land r) \lor p \) is true because \( p \lor (\text{anything}) \) is true.
   - \( V \Rightarrow V \) is true.

   **Answer**: Verdadero (True)

- **5.3**: \((r \land \neg q) \Leftrightarrow (q \Rightarrow p) \lor r\)

   - \( r \land \neg q = F \land F = F \).
   - \( q \Rightarrow p \) is true, so \( (q \Rightarrow p) \lor r = V \lor F = V \).
   - \( F \Leftrightarrow V \) is false.

   **Answer**: Falso (False)

- **5.4**: \((p \Rightarrow r) \Leftrightarrow (\neg q \land r) \Rightarrow \neg r\)

   - \( p \Rightarrow r = V \Rightarrow F = F \).
   - \( \neg q \land r = F \land F = F \), and \( F \Rightarrow \neg r = F \Rightarrow V = V \).
   - \( F \Leftrightarrow V \) is false.

   **Answer**: Falso (False)

Do you have any questions or would you like details about any part of the solution? Here are 5 questions to deepen your understanding:

1. How do truth tables work to verify propositions?
2. What is the difference between \( \Rightarrow \) (implication) and \( \Leftrightarrow \) (biconditional)?
3. Can the truth value of a proposition be independent of the truth values of its components?
4. Why does \( p \Rightarrow r \) evaluate to false when \( p \) is true and \( r \) is false?
5. How does the truth of \( \neg p \lor q \) change if both \( p \) and \( q \) are false?

**Tip**: In logical operations, pay attention to the distinction between "and" ( \( \land \) ) and "or" ( \( \lor \) ). This changes the way propositions combine their truth values.

Determine the truth value of the logical propositions based on given values: p is True, q is True, and r is False.

Determine the truth values of logical propositions involving p, q, and r, with p = True, q = True, and r = False. This problem covers key concepts in propositional logic, including implication, conjunction, disjunction, negation, and biconditional connectives.

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