Use only the inference rules MP, MT, DS and HS for this quiz. For each proof, you must include (i.e., write) the premises in that proof. Proofs without premises will lead to points being lost. YOU CANNOT USE CONDITIONAL PROOF (CP), INDIRECT PROOF (IP), OR ASSUMED PREMISES (AP). You will lose points if you use any of the inference rules Resolution, Contradiction, Transposition, Idempotence and Identity. Here is the transcription of the text from the image that can be used for an educational website:---### Logical Proof ExercisesThe following exercises provide practice in constructing logical proofs. For each set of premises, prove the given conclusion.#### Exercise 1- **Premises:** 1. \( C \supset (\sim D \supset \sim X) \) 2. \( C \)- **Conclusion:** 3. \( X \)**Goal:** Prove \( D \) **Note:** /D means to prove D.---#### Exercise 2- **Premises:** 1. \( A \supset \sim B \) 2. \( B \)- **Conclusion:** 3. \( \sim A \supset (C \lor \sim B) \)**Goal:** Prove \( C \)---#### Exercise 3- **Premises:** 1. \( A \supset \sim D \)- **Conclusion:** 2. \( \sim D \supset (\sim B \lor \sim \sim A) \) 3. \( A \) 4. \( B \)**Goal:** Prove \( \sim \sim A \)---These exercises involve using logical connectives such as implication (\(\supset\)), negation (\(\sim\)), and disjunction (\(\lor\)). Work through each proof step by step to arrive at the desired conclusion.

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Use only the inference rules MP, MT, DS and HS for this quiz. For each proof, you must include (i.e., write) the premises in that proof.

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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For each proof, you must include (i.e., write) the premises in that proof. Proofs must have premises. Conditional Proof (CP), Indirect Proof (IP) and Assumed Premises (AP) are not allowed to be used. Only the 18 rules of inference can be used. Points will be lost if you use any of the inference rules Resolution, Contradiction, Transposition, Idempotence and Identity. Only answer questions 7, 9 and 11.