**Consider the F-L Axioms:**- **Axiom 1.** $ A \rightarrow (B \rightarrow A) $- **Axiom 2.** $ (A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C)) $- **Axiom 3.** $ (\neg A \rightarrow \neg B) \rightarrow (B \rightarrow A) $**Proof rules allowed:** MP (Modus Ponens), CP (Conditional Proof), HS (Hypothetical Syllogism; last two proved in class)---Consider the following proof for:\[ A \rightarrow \neg A \]**Proof:**1. $ A $2. $ \neg A \rightarrow (\neg \neg A \rightarrow \neg A) $3. $ \neg \neg A \rightarrow \neg A $4. $ (\neg \neg A \rightarrow \neg A) \rightarrow (A \rightarrow \neg A) $5. $ A \rightarrow \neg A $---**Which of the following is most accurate?**- ○ Correct proof- ○ Incorrect at line 2- ○ Incorrect at line 3- ○ Incorrect at line 4 The following proof for the tautology \[(A \land B \to C) \to (A \to (B \to C))\]contains two nested sub-proofs. What line numbers correspond to the two sub-proofs?**Proof:**1. $ (A \land B) \to C $ $ P $2. $ A $ $ P $3. $ B $ $ P $4. $ A \land B $ $ 2, 3 $ Conj5. $ C $ $ 1, 4 $ MP6. $ B \to C $ $ 3, 5 $ CP7. $ A \to (B \to C) $ $ 2, 6 $ CP QED $ 1, 7 $ CP**Options:**- ○ Lines 2-6 and lines 4-6- ○ Lines 2-6 and lines 3-5- ○ Lines 3-6 and lines 3-5- ○ Lines 3-6 and lines 4-5

Axiom 1. A → (B → A) Axiom 2. (A → (B → C)) → ((A → B) → (A → C)) Axiom 3. (¬A → ¬B) → (B → A) Proof rules allowed: MP, CP, HS (last two proved in class) Consider the following proof for: A → ¬A Proof: 1.A 2.¬A → (¬¬A → ¬A) 3.¬¬A → ¬A 4.(¬¬A → ¬A) → (A → ¬A) 5.A → ¬A Which of the following is most accurate? Correct proof Incorrect at line 2 Incorrect at line 3 Incorrect at line 4

Axiom 1. A → (B → A) Axiom 2. (A → (B → C)) → ((A → B) → (A → C)) Axiom 3. (¬A → ¬B) → (B → A) Proof rules allowed: MP, CP, HS (last two proved in class) Consider the following proof for: A → ¬A Proof: 1.A 2.¬A → (¬¬A → ¬A) 3.¬¬A → ¬A 4.(¬¬A → ¬A) → (A → ¬A) 5.A → ¬A Which of the following is most accurate? Correct proof Incorrect at line 2 Incorrect at line 3 Incorrect at line 4

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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Question

**Consider the F-L Axioms:**

- **Axiom 1.** $ A \rightarrow (B \rightarrow A) $
- **Axiom 2.** $ (A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C)) $
- **Axiom 3.** $ (\neg A \rightarrow \neg B) \rightarrow (B \rightarrow A) $

**Proof rules allowed:** MP (Modus Ponens), CP (Conditional Proof), HS (Hypothetical Syllogism; last two proved in class)

---

Consider the following proof for:

\[ A \rightarrow \neg A \]

**Proof:**

1. $ A $
2. $ \neg A \rightarrow (\neg \neg A \rightarrow \neg A) $
3. $ \neg \neg A \rightarrow \neg A $
4. $ (\neg \neg A \rightarrow \neg A) \rightarrow (A \rightarrow \neg A) $
5. $ A \rightarrow \neg A $

---

**Which of the following is most accurate?**

- ○ Correct proof
- ○ Incorrect at line 2
- ○ Incorrect at line 3
- ○ Incorrect at line 4

The following proof for the tautology

\[
(A \land B \to C) \to (A \to (B \to C))
\]

contains two nested sub-proofs. What line numbers correspond to the two sub-proofs?

**Proof:**

1. $ (A \land B) \to C $ $ P $

2. $ A $ $ P $

3. $ B $ $ P $

4. $ A \land B $ $ 2, 3 $ Conj

5. $ C $ $ 1, 4 $ MP

6. $ B \to C $ $ 3, 5 $ CP

7. $ A \to (B \to C) $ $ 2, 6 $ CP

QED $ 1, 7 $ CP

**Options:**

- ○ Lines 2-6 and lines 4-6
- ○ Lines 2-6 and lines 3-5
- ○ Lines 3-6 and lines 3-5
- ○ Lines 3-6 and lines 4-5

Expert Solution

Step 1

"Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question."

Consider the F-L Axioms:

Axiom 1. $A \to (B \to A)$

Axiom 2. $A \to (B \to C) \to ((A \to B) \to (A \to C))$

Axiom 3. $(~ A \to ~ B) \to (B \to A)$

Its proof is given by

$1 . A 2 . ~ A \to (~ ~ A \to ~ A) 3 . ~ ~ A \to ~ A 4 . (~ ~ A \to ~ A) \to (A \to ~ A) 5 . ~ A \to A$

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