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To address part III of your problem, we are tasked with proving that the logical statement:

\[
[(p \lor q) \land (p \to r) \land (q \to r)] \to r
\]

is a tautology using logical equivalences.

### Steps to Prove:

1. **Expand implications**:
   - The conditional statements \(p \to r\) and \(q \to r\) can be rewritten using their equivalent forms:
   \[
   p \to r \equiv \neg p \lor r
   \]
   \[
   q \to r \equiv \neg q \lor r
   \]
   So the given expression becomes:
   \[
   [(p \lor q) \land (\neg p \lor r) \land (\neg q \lor r)] \to r
   \]

2. **Assume the antecedent**:
   - We can now analyze the antecedent of the implication:
   \[
   (p \lor q) \land (\neg p \lor r) \land (\neg q \lor r)
   \]
   Break it down:
   - The first term \(p \lor q\) states that either \(p\) or \(q\) is true.
   - The second term \(\neg p \lor r\) states that either \(p\) is false or \(r\) is true.
   - The third term \(\neg q \lor r\) states that either \(q\) is false or \(r\) is true.

3. **Consider possible cases for \(p\) and \(q\)**:
   We will analyze the truth values of \(p\) and \(q\) and check if the expression always leads to \(r\) being true:
   - Case 1: If \(p\) is true and \(q\) is false, then \(\neg p\) is false, so \(r\) must be true from the second condition \(\neg p \lor r\).
   - Case 2: If \(q\) is true and \(p\) is false, then \(\neg q\) is false, so \(r\) must be true from the third condition \(\neg q \lor r\).
   - Case 3: If both \(p\) and \(q\) are false, then \(r\) must still be true to satisfy both \(\neg p \lor r\) and \(\neg q \lor r\).

4. **Conclusion**:
   - In all possible cases for \(p\) and \(q\), the expression leads to \(r\) being true. Therefore, the whole statement is a tautology, because it is true regardless of the truth values of \(p\) and \(q\).

Thus, the statement \([(p \lor q) \land (p \to r) \land (q \to r)] \to r\) is a tautology.

Would you like further clarification on any part of this, or would you like to see the next part (IV) addressed?

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### Related Questions:
1. What is the definition of a tautology in propositional logic?
2. How can implications be rewritten using logical equivalences?
3. What are the truth values of a tautology under all possible conditions?
4. How does one apply truth tables to verify a tautology?
5. Can a similar method be used to prove other types of logical statements?

### Tip:
When working with logical implications, always remember that \(p \to q\) is equivalent to \(\neg p \lor q\). This can greatly simplify many problems involving conditionals.

Show that the conditional statement [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r is a tautology by applying a chain of logical equivalences.

Learn how to prove the statement [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r is a tautology using logical equivalences. This solution breaks down the proof step-by-step, making it suitable for advanced high school or early university students.

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