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The problem asks to determine the truth value of three logical propositions, given that \( p \) and \( r \) are true propositions and \( q \) is false.

We will evaluate each part step-by-step.

### Given:
- \( p \) is true (T)
- \( r \) is true (T)
- \( q \) is false (F)

### Proposition a) \[ (p \land \neg q) \lor \neg r \implies q \]

**Step 1: Evaluate \( p \land \neg q \)**  
- \( p \) is true, \( \neg q \) means "not \( q \)," so \( \neg q \) is true since \( q \) is false.  
- \( p \land \neg q = T \land T = T \).

**Step 2: Evaluate \( \neg r \)**  
- Since \( r \) is true, \( \neg r \) is false.

**Step 3: Evaluate \( (p \land \neg q) \lor \neg r \)**  
- This is \( T \lor F = T \).

**Step 4: Evaluate \( (p \land \neg q) \lor \neg r \implies q \)**  
- We have \( T \implies F \).  
- The implication \( T \implies F \) is false.

Thus, the truth value of **a) is false**.

---

### Proposition b) \[ (\neg r \lor q) \land (r \lor \neg p) \iff \neg \neg r \]

**Step 1: Evaluate \( \neg r \lor q \)**  
- \( \neg r \) is false, and \( q \) is false.  
- \( \neg r \lor q = F \lor F = F \).

**Step 2: Evaluate \( r \lor \neg p \)**  
- \( r \) is true, \( \neg p \) is false since \( p \) is true.  
- \( r \lor \neg p = T \lor F = T \).

**Step 3: Evaluate \( (\neg r \lor q) \land (r \lor \neg p) \)**  
- This is \( F \land T = F \).

**Step 4: Evaluate \( \neg \neg r \)**  
- \( \neg \neg r = r = T \).

**Step 5: Evaluate \( (\neg r \lor q) \land (r \lor \neg p) \iff \neg \neg r \)**  
- This is \( F \iff T \), which is false.

Thus, the truth value of **b) is false**.

---

### Proposition c) \[ (\neg p \implies q) \iff \neg r \lor (\neg q \implies r) \]

**Step 1: Evaluate \( \neg p \implies q \)**  
- \( \neg p \) is false, and \( q \) is false.  
- \( \neg p \implies q = F \implies F = T \).

**Step 2: Evaluate \( \neg q \implies r \)**  
- \( \neg q \) is true, and \( r \) is true.  
- \( \neg q \implies r = T \implies T = T \).

**Step 3: Evaluate \( \neg r \lor (\neg q \implies r) \)**  
- \( \neg r \) is false, and \( \neg q \implies r \) is true.  
- \( \neg r \lor (\neg q \implies r) = F \lor T = T \).

**Step 4: Evaluate \( (\neg p \implies q) \iff \neg r \lor (\neg q \implies r) \)**  
- This is \( T \iff T \), which is true.

Thus, the truth value of **c) is true**.

---

### Final Answers:
- a) False
- b) False
- c) True

Would you like more details or further clarification? Here are five related questions:

1. How is the truth table for implications derived?
2. What is the significance of logical connectives like AND, OR, and NOT?
3. How does the biconditional (iff) operator work in logical statements?
4. How do truth tables help verify logical expressions?
5. What is the difference between material implication and logical equivalence?

**Tip:** In logical implications, a false antecedent always makes the implication true!

Si p y r son proposiciones verdaderas y q es falsa, determine el valor de verdad de: a) [(p ∧ ¬q) ∨ ¬r] ⇒ q, b) [(¬r ∨ q) ∧ (r ∨ ¬p)] ⇔ ¬¬r, c) [(¬p ⇒ q) ⇔ ¬r ∨ (¬q ⇒ r)]

Determine the truth value of three logical propositions given p and r as true and q as false. Explore logical connectives such as implication, biconditional, conjunction, and negation, with a detailed step-by-step solution.

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