**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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