Key Concepts and Formulas

• Implication: $p \to q \equiv \sim p \lor q$
• De Morgan's Laws: $\sim (p \land q) \equiv \sim p \lor \sim q$ and $\sim (p \lor q) \equiv \sim p \land \sim q$
• Tautology: A statement that is always true, regardless of the truth values of its components. $p \lor \sim p$ is a tautology.

Step-by-Step Solution

Step 1: Rewrite the implication using the equivalence $p \to q \equiv \sim p \lor q$. We are given the expression $(p \wedge (p \to q) \wedge (q \to r)) \to r$. We replace $p \to q$ with $\sim p \lor q$ and $q \to r$ with $\sim q \lor r$. $(p \wedge (p \to q) \wedge (q \to r)) \to r \equiv (p \wedge (\sim p \lor q) \wedge (\sim q \lor r)) \to r$

Step 2: Simplify the expression inside the parentheses using the distributive property and associative property. We will first simplify $p \wedge (\sim p \lor q)$. Using the distributive property, $p \wedge (\sim p \lor q) \equiv (p \wedge \sim p) \lor (p \wedge q)$. Since $p \wedge \sim p$ is always false, we have $(p \wedge \sim p) \lor (p \wedge q) \equiv F \lor (p \wedge q) \equiv p \wedge q$. Therefore, our expression becomes: $(p \wedge q \wedge (\sim q \lor r)) \to r$

Step 3: Simplify further using the distributive property. Now we simplify $(p \wedge q) \wedge (\sim q \lor r)$. Using the distributive property, we get $(p \wedge q \wedge \sim q) \lor (p \wedge q \wedge r)$. Since $q \wedge \sim q$ is always false, $p \wedge q \wedge \sim q$ is also always false. Thus, $(p \wedge q \wedge \sim q) \lor (p \wedge q \wedge r) \equiv F \lor (p \wedge q \wedge r) \equiv p \wedge q \wedge r$. Our expression is now: $(p \wedge q \wedge r) \to r$

Step 4: Rewrite the implication using the equivalence $p \to q \equiv \sim p \lor q$. $(p \wedge q \wedge r) \to r \equiv \sim (p \wedge q \wedge r) \lor r$

Step 5: Apply De Morgan's Law to simplify the negation. Using De Morgan's Law, $\sim (p \wedge q \wedge r) \equiv \sim p \lor \sim q \lor \sim r$. So we have: $\sim p \lor \sim q \lor \sim r \lor r$

Step 6: Recognize the tautology. Since $r \lor \sim r$ is always true, the entire expression $\sim p \lor \sim q \lor \sim r \lor r$ is always true, regardless of the truth values of $p$ and $q$. Therefore, the expression is a tautology. $\sim p \lor \sim q \lor (\sim r \lor r) \equiv \sim p \lor \sim q \lor T \equiv T$

Common Mistakes & Tips

• When simplifying logical expressions, it is crucial to remember the order of operations and the equivalences of logical operators.
• De Morgan's Laws and the implication equivalence are key tools for simplifying complex expressions.
• Recognizing patterns that lead to tautologies or contradictions can save significant time.

Summary

We started with the given expression and systematically simplified it using logical equivalences. We converted implications to disjunctions, applied De Morgan's laws, and used the distributive property. Eventually, we arrived at an expression that contained $r \lor \sim r$, which is always true, making the entire expression a tautology.

Final Answer

The final answer is \boxed{a tautology}, which corresponds to option (A).