https://file.aiquickdraw.com/mathbot/file/1724574468871eu6bdlaw.pdf

It seems that the document contains a list of logical problems and exercises related to topics such as logical equivalence, tautologies, contradictions, inference, predicate logic, and automated theorem proving. Let me go ahead and solve the first problem listed in the document:

**Problem 26:** Show that ¬p → (q → r) and q → (p ∨ r) are logically equivalent.

### Step-by-Step Solution:

1. **Expression 1:** ¬p → (q → r)

   The first expression can be expanded using the implication rule \( p \rightarrow q \equiv \neg p \vee q \):

   \[
   \neg p \rightarrow (q \rightarrow r) \equiv \neg (\neg p) \vee (\neg q \vee r)
   \]
   Simplifying further:

   \[
   p \vee (\neg q \vee r)
   \]
   Which can be rewritten as:

   \[
   p \vee \neg q \vee r
   \]

2. **Expression 2:** q → (p ∨ r)

   Similarly, expand using the implication rule:

   \[
   q \rightarrow (p \vee r) \equiv \neg q \vee (p \vee r)
   \]

   Which simplifies to:

   \[
   \neg q \vee p \vee r
   \]

3. **Comparison:**

   Now, comparing both expressions:

   - \( p \vee \neg q \vee r \) (from Expression 1)
   - \( \neg q \vee p \vee r \) (from Expression 2)

   Both expressions are the same. Therefore:

   \[
   \neg p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \vee r)
   \]

Hence, the two logical expressions are indeed **logically equivalent**.

---

Would you like a detailed explanation of any step, or do you have any questions?

### 5 Related Questions:

1. How can De Morgan's laws be used to simplify logical expressions?
2. What is the difference between a tautology and a contradiction in logic?
3. How do you prove logical equivalence using truth tables?
4. What is the significance of logical equivalence in automated theorem proving?
5. Can two non-equivalent logical expressions produce the same truth table under certain conditions?

### Tip:
When working with logical equivalences, always remember to apply implication, De Morgan's laws, and distribution systematically. This ensures that the logical expressions are simplified correctly and compared accurately.

Explore the logical equivalence proof between ¬p → (q → r) and q → (p ∨ r) with detailed step-by-step solutions. This problem delves into logical equivalence, implications, and the application of De Morgan's laws in logic.

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