https://file.aiquickdraw.com/mathbot/file/1724587035013lr9zu198.pdf

The assignment file contains a set of problems on topics related to logic, predicate logic, and automated theorem proving. Let's solve one of the problems mentioned.

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

#### Solution:

To prove that ¬p → (q → r) and q → (p ∨ r) are logically equivalent, we'll use truth tables.

Let's break down both expressions:

1. **Expression 1: ¬p → (q → r)**
   - The implication ¬p → (q → r) means "If ¬p is true, then q → r must be true."
   - The truth table for q → r is:  
     - True when q is False or r is True.

2. **Expression 2: q → (p ∨ r)**
   - The implication q → (p ∨ r) means "If q is true, then p ∨ r must be true."
   - The truth table for p ∨ r is:  
     - True when at least one of p or r is true.

Let's now construct the truth tables:

| p | q | r | ¬p | q → r | ¬p → (q → r) | p ∨ r | q → (p ∨ r) |
|---|---|---|----|-------|--------------|-------|-------------|
| T | T | T |  F |   T   |      T       |   T   |     T       |
| T | T | F |  F |   F   |      T       |   T   |     T       |
| T | F | T |  F |   T   |      T       |   T   |     T       |
| T | F | F |  F |   T   |      T       |   F   |     T       |
| F | T | T |  T |   T   |      T       |   T   |     T       |
| F | T | F |  T |   F   |      F       |   F   |     F       |
| F | F | T |  T |   T   |      T       |   T   |     T       |
| F | F | F |  T |   T   |      T       |   F   |     T       |

#### Analysis:
The columns for ¬p → (q → r) and q → (p ∨ r) are identical, meaning these two expressions are logically equivalent.

#### Conclusion:
Therefore, ¬p → (q → r) and q → (p ∨ r) are indeed logically equivalent.

Would you like any further details or have any questions? 

Here are some related problems for practice:

1. Prove the logical equivalence between ¬(p ∨ q) and ¬p ∧ ¬q.
2. Show that (p ∧ q) → r is logically equivalent to p → (q → r).
3. Determine if ¬(p ∧ q) and ¬p ∨ ¬q are logically equivalent.
4. Prove that p → q is equivalent to ¬p ∨ q.
5. Show that ¬(p → q) is equivalent to p ∧ ¬q.

**Tip:** When proving logical equivalences, truth tables are a straightforward method to verify the results, but algebraic manipulations using logical identities can provide deeper insight.

Explore the logical equivalence proof between ¬p → (q → r) and q → (p ∨ r) using truth tables and logical analysis. This problem delves into predicate logic and implications, suitable for advanced level learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation