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?

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.