We are tasked with determining whether the logical equivalence below holds:

$p \to (q \to r) \leftrightarrow (p \land q) \to r$

Step 1: Simplify each side of the biconditional statement.

Left-hand side: $p \to (q \to r)$

This expression represents an implication where $p$ implies another implication, $q \to r$.

We can rewrite $q \to r$ as $\neg q \lor r$. Therefore, the entire left-hand side becomes: $p \to (\neg q \lor r)$ Using the material implication rule $p \to A \equiv \neg p \lor A$, we rewrite $p \to (\neg q \lor r)$ as: $\neg p \lor (\neg q \lor r)$ This simplifies to: $\neg p \lor \neg q \lor r$

Right-hand side: $(p \land q) \to r$

Again using the material implication rule, we rewrite $(p \land q) \to r$ as: $\neg (p \land q) \lor r$ We know that $\neg (p \land q)$ is logically equivalent to $\neg p \lor \neg q$, so this becomes: $(\neg p \lor \neg q) \lor r$ which simplifies to: $\neg p \lor \neg q \lor r$

Step 2: Compare the two sides

The left-hand side simplifies to $\neg p \lor \neg q \lor r$, and the right-hand side also simplifies to $\neg p \lor \neg q \lor r$.

Since both sides are equivalent, we conclude that the original statement is true: $p \to (q \to r) \leftrightarrow (p \land q) \to r$

Conclusion

The given statement is a tautology, meaning it holds true for all truth values of $p$, $q$, and $r$.

Would you like more details on the steps or have any further questions?

Here are some related questions for further practice:

1. What is the truth table for $p \to (q \to r)$?
2. How can De Morgan's laws help simplify logical expressions?
3. How is the material implication rule applied in propositional logic?
4. What is the contrapositive of $p \to (q \to r)$?
5. Can you prove this equivalence using truth tables instead of symbolic manipulation?

Tip: Always check if both sides of a logical equivalence simplify to the same form, as it can make the verification much easier!