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For each of the following two proofs, outline the proof and fill in any details that will make the proof easier to understand. Answer the following for each proof. a. What is the goal of the proof? b. What is the hypothesis? c. What definitions are necessary? d. What axioms, previously proved facts, or laws of logic are used in the proof? i. Let A and B be arbitrary sets. Prove: B- (B-A) ≤ B Proof. Let x E B - (B-A). Then x E B and x B-A. Hence, x E B. ii. Let x and y be integers. Prove: If x and y are odd, then x + y is even. Proof. Assume x and y are odd integers. Then x = 2j + 1 and y = 2k + 1 for some integers j and k. Hence, x + y = 2j + 1 + 2k + 1 = 2j + 2k + 2 = 2(j + k + 1). Therefore, x + y is an even integer.

For each of the following two proofs, outline the proof and fill in any details that will make the proof easier to understand. Answer the following for each proof. a. What is the goal of the proof? b. What is the hypothesis? c. What definitions are necessary? d. What axioms, previously proved facts, or laws of logic are used in the proof? i. Let A and B be arbitrary sets. Prove: B- (B-A) ≤ B Proof. Let x E B - (B-A). Then x E B and x B-A. Hence, x E B. ii. Let x and y be integers. Prove: If x and y are odd, then x + y is even. Proof. Assume x and y are odd integers. Then x = 2j + 1 and y = 2k + 1 for some integers j and k. Hence, x + y = 2j + 1 + 2k + 1 = 2j + 2k + 2 = 2(j + k + 1). Therefore, x + y is an even integer.

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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For each of the following two proofs, outline the proof and fill in any details that will make the proof easier to understand. Answer the following for each proof. a. What is the goal of the proof? b. What is the hypothesis? c. What definitions are necessary? d. What axioms, previously proved facts, or laws of logic are used in the proof? i. Let A and B be arbitrary sets. Prove: B- (B-A) ≤ B Proof. Let x E B - (BA). Then x E B and x & B-A. Hence, x E B. ii. Let x and y be integers. Prove: If x and y are odd, then x + y is even. Proof. Assume x and y are odd integers. Then x = 2j + 1 and y = 2k + 1 for some integers j and k. Hence, x+y=2j+ 1 + 2k + 1 = 2j + 2k + 2 = 2(j + k + 1). Therefore, x + y is an even integer.
Transcribed Image Text:For each of the following two proofs, outline the proof and fill in any details that will make the proof easier to understand. Answer the following for each proof. a. What is the goal of the proof? b. What is the hypothesis? c. What definitions are necessary? d. What axioms, previously proved facts, or laws of logic are used in the proof? i. Let A and B be arbitrary sets. Prove: B- (B-A) ≤ B Proof. Let x E B - (BA). Then x E B and x & B-A. Hence, x E B. ii. Let x and y be integers. Prove: If x and y are odd, then x + y is even. Proof. Assume x and y are odd integers. Then x = 2j + 1 and y = 2k + 1 for some integers j and k. Hence, x+y=2j+ 1 + 2k + 1 = 2j + 2k + 2 = 2(j + k + 1). Therefore, x + y is an even integer.
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