Determine whether the following are valid deduction rules. Your answer should involve a truth table as well as an explanation of what your truth table tells you. (a) (b) P→ (QVR) ¬(P→Q) R (PAQ) → R ¬PV¬Q ¬R .. ..
Determine whether the following are valid deduction rules. Your answer should involve a truth table as well as an explanation of what your truth table tells you. (a) (b) P→ (QVR) ¬(P→Q) R (PAQ) → R ¬PV¬Q ¬R .. ..
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**3. Determine whether the following are valid deduction rules. Your answer should involve a truth table as well as an explanation of what your truth table tells you.**
__(a)__
\[ P \rightarrow (Q \vee R) \]
\[ \neg (P \rightarrow Q) \]
\[\therefore R \]
__(b)__
\[ (P \wedge Q) \rightarrow R \]
\[ \neg P \vee \neg Q \]
\[\therefore \neg R \]
**Explanation:**
To determine whether these deduction rules are valid, we will utilize truth tables. A deduction rule is valid if the premises logically lead to the conclusion in every scenario.
### (a) Truth Table
| \(P\) | \(Q\) | \(R\) | \(P \rightarrow (Q \vee R)\) | \(P \rightarrow Q\) | \(\neg (P \rightarrow Q)\) | Conclusion \(R\) |
|------|------|------|-----------------------------|----------------|-------------------------|--------------|
| T | T | T | T | T | F | T |
| T | T | F | T | T | F | F |
| T | F | T | T | F | T | T |
| T | F | F | F | F | T | F |
| F | T | T | T | T | F | T |
| F | T | F | T | T | F | F |
| F | F | T | T | F | T | T |
| F | F | F | T | F | T | F |
By analyzing the truth table, we can conclude that there are situations where the premises are true and the conclusion \(R\) is false (rows 2, 4, 6, 8). Thus, this deduction rule is not valid.
###](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2F98d1ebfa-1bbe-42e9-a6fe-c7297d697d35%2Fhnc36hh_processed.png&w=3840&q=75)
Transcribed Image Text:**3. Determine whether the following are valid deduction rules. Your answer should involve a truth table as well as an explanation of what your truth table tells you.**
__(a)__
\[ P \rightarrow (Q \vee R) \]
\[ \neg (P \rightarrow Q) \]
\[\therefore R \]
__(b)__
\[ (P \wedge Q) \rightarrow R \]
\[ \neg P \vee \neg Q \]
\[\therefore \neg R \]
**Explanation:**
To determine whether these deduction rules are valid, we will utilize truth tables. A deduction rule is valid if the premises logically lead to the conclusion in every scenario.
### (a) Truth Table
| \(P\) | \(Q\) | \(R\) | \(P \rightarrow (Q \vee R)\) | \(P \rightarrow Q\) | \(\neg (P \rightarrow Q)\) | Conclusion \(R\) |
|------|------|------|-----------------------------|----------------|-------------------------|--------------|
| T | T | T | T | T | F | T |
| T | T | F | T | T | F | F |
| T | F | T | T | F | T | T |
| T | F | F | F | F | T | F |
| F | T | T | T | T | F | T |
| F | T | F | T | T | F | F |
| F | F | T | T | F | T | T |
| F | F | F | T | F | T | F |
By analyzing the truth table, we can conclude that there are situations where the premises are true and the conclusion \(R\) is false (rows 2, 4, 6, 8). Thus, this deduction rule is not valid.
###
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