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.

Explore the logical equivalence between ¬p → (q → r) and q → (p ∨ r) using logical identities. This detailed analysis breaks down each step to prove their equivalence, essential for understanding logical reasoning. Suitable for advanced learners interested in logic and mathematics.

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