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A. Prove that if n is an odd integer, then 7n - 5 is even. Give three different proofs, using direct proof, proof by contrapositive, and proof by contradiction. B. Let A and B be sets. For each statement below, determine if it is true or false. If true, prove the statement. If false, provide a counterexample. (i) A C B if and only if P(A) = P(B). (ii) P(A - B) = P(A) - P(B)

A. Prove that if n is an odd integer, then 7n - 5 is even. Give three different proofs, using direct proof, proof by contrapositive, and proof by contradiction. B. Let A and B be sets. For each statement below, determine if it is true or false. If true, prove the statement. If false, provide a counterexample. (i) A C B if and only if P(A) = P(B). (ii) P(A - B) = P(A) - P(B)

Advanced Engineering Mathematics
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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Please solve the following problem please (this is a two part question) 

**A.** Prove that if \( n \) is an odd integer, then \( 7n - 5 \) is even. Give three different proofs, using direct proof, proof by contrapositive, and proof by contradiction. **B.** Let \( A \) and \( B \) be sets. For each statement below, determine if it is true or false. If true, prove the statement. If false, provide a counterexample. (i) \( A \subseteq B \) if and only if \( P(A) \subseteq P(B) \). (ii) \( P(A - B) = P(A) - P(B) \).
Transcribed Image Text:**A.** Prove that if \( n \) is an odd integer, then \( 7n - 5 \) is even. Give three different proofs, using direct proof, proof by contrapositive, and proof by contradiction. **B.** Let \( A \) and \( B \) be sets. For each statement below, determine if it is true or false. If true, prove the statement. If false, provide a counterexample. (i) \( A \subseteq B \) if and only if \( P(A) \subseteq P(B) \). (ii) \( P(A - B) = P(A) - P(B) \).
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