1. (10 points) Using a truth table, determine if ((p → q) → r) and (p (q → r)) are logically equivalent.
1. (10 points) Using a truth table, determine if ((p → q) → r) and (p (q → r)) are logically equivalent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Question 1: Logical Equivalence Through Truth Tables**
**(10 points)** Using a truth table, determine if \(\left( (p \rightarrow q) \rightarrow r \right)\) and \(\left( p \rightarrow (q \rightarrow r) \right)\) are logically equivalent.
To solve this problem, create a truth table that evaluates the truth values of both logical expressions for all possible truth values of \(p\), \(q\), and \(r\). Then, compare the resulting columns of the truth table to check for logical equivalence.
**Steps to Create the Truth Table:**
1. **List all possible combinations of truth values for \(p\), \(q\), and \(r\).**
2. **Calculate \(p \rightarrow q\) for each combination.**
3. **Calculate \((p \rightarrow q) \rightarrow r\) for each combination.**
4. **Calculate \(q \rightarrow r\) for each combination.**
5. **Calculate \(p \rightarrow (q \rightarrow r)\) for each combination.**
6. **Compare the resulting columns from steps 3 and 5 to determine if they match in all cases.**
The expressions \(\left( (p \rightarrow q) \rightarrow r \right)\) and \(\left( p \rightarrow (q \rightarrow r) \right)\) are logically equivalent if and only if the columns for these two expressions in the truth table are identical.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d68e090-7e11-4ab8-a19a-f8ec202a32a2%2F930c0d6f-4ab0-41cd-9242-c6c8f5851d2a%2Frkklobb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 1: Logical Equivalence Through Truth Tables**
**(10 points)** Using a truth table, determine if \(\left( (p \rightarrow q) \rightarrow r \right)\) and \(\left( p \rightarrow (q \rightarrow r) \right)\) are logically equivalent.
To solve this problem, create a truth table that evaluates the truth values of both logical expressions for all possible truth values of \(p\), \(q\), and \(r\). Then, compare the resulting columns of the truth table to check for logical equivalence.
**Steps to Create the Truth Table:**
1. **List all possible combinations of truth values for \(p\), \(q\), and \(r\).**
2. **Calculate \(p \rightarrow q\) for each combination.**
3. **Calculate \((p \rightarrow q) \rightarrow r\) for each combination.**
4. **Calculate \(q \rightarrow r\) for each combination.**
5. **Calculate \(p \rightarrow (q \rightarrow r)\) for each combination.**
6. **Compare the resulting columns from steps 3 and 5 to determine if they match in all cases.**
The expressions \(\left( (p \rightarrow q) \rightarrow r \right)\) and \(\left( p \rightarrow (q \rightarrow r) \right)\) are logically equivalent if and only if the columns for these two expressions in the truth table are identical.
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