Consider the following proof of that IS supposed to prove the tautology: (A v B → C^ D) → (B → D) Proof: 1. AVВ — СлD P 2. ¬(B → D) P for IP 3. B^ ¬D 2 4. B 3, ?
Consider the following proof of that IS supposed to prove the tautology: (A v B → C^ D) → (B → D) Proof: 1. AVВ — СлD P 2. ¬(B → D) P for IP 3. B^ ¬D 2 4. B 3, ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Consider the following proof that is supposed to prove the tautology:
\[
(A \lor B \to C \land D) \to (B \to D)
\]
**Proof:**
1. \( A \lor B \to C \land D \) \hspace{2em} \( P \)
2. \( \neg(B \to D) \) \hspace{2em} \( P \) for IP
3. \( B \land \neg D \) \hspace{2em} 2
4. \( B \) \hspace{2em} 3, ?
5. \( A \lor B \) \hspace{2em} 4, ?
6. \( C \land D \) \hspace{2em} 1, 5, ?
7. \( \neg D \) \hspace{2em} 3, ?
8. \( D \) \hspace{2em} 6, ?
9. \( D \land \neg D \) \hspace{2em} 7, 8, ?
10. False \hspace{2em} 9, Contr
**QED: 1, 2, 10, IP**
The above proof is correct. However, in lines 4 -- 9, we have not indicated the Proof Rule being used (indicated by a `?`). Which of the following choices gives the correct sequence of Proof Rules for these lines?
- ○ Add, Add, Conj, DN, Contr, Conj
- ○ Simp, Add, MP, Simp, Simp, Conj
- ● Simp, Simp, Add, Simp, DN, Contr
- ○ Add, Simp, Simp, Simp, Simp, Contr](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff43fe492-0086-4aac-90eb-2b796e708e5e%2F80b8a58f-8540-4797-9ae8-ff99bf44e066%2Fsko973p_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following proof that is supposed to prove the tautology:
\[
(A \lor B \to C \land D) \to (B \to D)
\]
**Proof:**
1. \( A \lor B \to C \land D \) \hspace{2em} \( P \)
2. \( \neg(B \to D) \) \hspace{2em} \( P \) for IP
3. \( B \land \neg D \) \hspace{2em} 2
4. \( B \) \hspace{2em} 3, ?
5. \( A \lor B \) \hspace{2em} 4, ?
6. \( C \land D \) \hspace{2em} 1, 5, ?
7. \( \neg D \) \hspace{2em} 3, ?
8. \( D \) \hspace{2em} 6, ?
9. \( D \land \neg D \) \hspace{2em} 7, 8, ?
10. False \hspace{2em} 9, Contr
**QED: 1, 2, 10, IP**
The above proof is correct. However, in lines 4 -- 9, we have not indicated the Proof Rule being used (indicated by a `?`). Which of the following choices gives the correct sequence of Proof Rules for these lines?
- ○ Add, Add, Conj, DN, Contr, Conj
- ○ Simp, Add, MP, Simp, Simp, Conj
- ● Simp, Simp, Add, Simp, DN, Contr
- ○ Add, Simp, Simp, Simp, Simp, Contr
![Consider the following proof of the tautology:
\[
(A \rightarrow C) \rightarrow (A \land B \rightarrow C)
\]
Proof:
1. \( A \rightarrow C \)
2. \( A \land B \)
3. \( A \)
4. \( C \)
5. \( A \land B \rightarrow C \)
QED
This proof is an example of applying:
- [ ] Neither IP nor CP Rules
- [ ] Both CP and IP Rules
- [ ] IP Rule
- [x] CP Rule
### Explanation
The proof shows a logical deduction where the implication \( (A \rightarrow C) \rightarrow (A \land B \rightarrow C) \) is verified. It uses the assumption from line 2, \( A \land B \), and the given \( A \rightarrow C \) to arrive at the conclusion \( A \land B \rightarrow C \).
This uses the Conditional Proof (CP) Rule:
- The proof assumes \( A \land B \) to establish \( A \land B \rightarrow C \) using previous deductions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff43fe492-0086-4aac-90eb-2b796e708e5e%2F80b8a58f-8540-4797-9ae8-ff99bf44e066%2F5r4t788_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following proof of the tautology:
\[
(A \rightarrow C) \rightarrow (A \land B \rightarrow C)
\]
Proof:
1. \( A \rightarrow C \)
2. \( A \land B \)
3. \( A \)
4. \( C \)
5. \( A \land B \rightarrow C \)
QED
This proof is an example of applying:
- [ ] Neither IP nor CP Rules
- [ ] Both CP and IP Rules
- [ ] IP Rule
- [x] CP Rule
### Explanation
The proof shows a logical deduction where the implication \( (A \rightarrow C) \rightarrow (A \land B \rightarrow C) \) is verified. It uses the assumption from line 2, \( A \land B \), and the given \( A \rightarrow C \) to arrive at the conclusion \( A \land B \rightarrow C \).
This uses the Conditional Proof (CP) Rule:
- The proof assumes \( A \land B \) to establish \( A \land B \rightarrow C \) using previous deductions.
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