Let's solve the problem using logical identities instead of a truth table.

Problem 26: Logical Equivalence

Statement: Show that ¬p → (q → r) and q → (p ∨ r) are logically equivalent.

Solution Using Logical Identities:

We need to prove that: $\neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \vee r)$

Let's break this down step by step.

1. Express the implications using logical equivalences:

   • $A \rightarrow B$ is equivalent to $\neg A \vee B$.

   So, the first expression becomes: $\neg p \rightarrow (q \rightarrow r) \equiv \neg (\neg p) \vee (q \rightarrow r)$

   Simplifying the double negation: $p \vee (q \rightarrow r)$

   Now, express $q \rightarrow r$ using the equivalence $q \rightarrow r \equiv \neg q \vee r$: $p \vee (\neg q \vee r)$

   This can be rewritten using the associative and commutative properties of $\vee$ as: $(p \vee \neg q) \vee r$

2. Simplify the expression: We now have: $p \vee \neg q \vee r$

   On the other side, express the second statement $q \rightarrow (p \vee r)$ using logical identities: $q \rightarrow (p \vee r) \equiv \neg q \vee (p \vee r)$

   Using the commutative property of $\vee$: $\neg q \vee p \vee r$

3. Compare the expressions: Both expressions simplify to: $p \vee \neg q \vee r$

   Since both sides reduce to the same expression, the original statements are logically equivalent.

Conclusion:

Therefore, $\neg p \rightarrow (q \rightarrow r)$ and $q \rightarrow (p \vee r)$ are logically equivalent by using logical identities.

Would you like any further details or have any questions?

Here are some related problems for practice:

1. Prove that $p \vee \neg p$ is a tautology using logical identities.
2. Show that $\neg(p \vee q) \equiv \neg p \wedge \neg q$.
3. Prove the equivalence $(p \rightarrow q) \rightarrow r \equiv p \rightarrow (q \rightarrow r)$.
4. Demonstrate that $p \wedge \neg p$ is a contradiction.
5. Simplify $(p \vee q) \wedge (\neg p \vee q)$ using logical identities.

Tip: Understanding the basic logical identities, such as De Morgan's laws and the distributive, associative, and commutative properties, is crucial for simplifying and proving logical equivalences.