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(see instructions for Exercise 6 in Section 2.2): PROOF EVALUATION a) "THEOREM": For any sets A, B, and C, if A UB=AU C, then B = C. "Proof": We will prove this by contradiction. Suppose that A UB and A UC are not equal. Then there is some object x that is in one and not the other. We proceed by looking at two cases: First look at the case where xEA UB and not in A U C. Then x is not in A (be- cause, if it were, it would also be in A U C). So x must be in B (otherwise it wouldn't be in A U B). Also, x is not in C (because, if it were, it would also be in A U C). Therefore x is in B and not in C, which contradicts the condition B = C. The case where x EAU C and not in A U B is done by a parallel argument.

(see instructions for Exercise 6 in Section 2.2): PROOF EVALUATION a) "THEOREM": For any sets A, B, and C, if A UB=AU C, then B = C. "Proof": We will prove this by contradiction. Suppose that A UB and A UC are not equal. Then there is some object x that is in one and not the other. We proceed by looking at two cases: First look at the case where xEA UB and not in A U C. Then x is not in A (be- cause, if it were, it would also be in A U C). So x must be in B (otherwise it wouldn't be in A U B). Also, x is not in C (because, if it were, it would also be in A U C). Therefore x is in B and not in C, which contradicts the condition B = C. The case where x EAU C and not in A U B is done by a parallel argument.

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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(see instructions for Exercise 6 in Section 2.2): PROOF EVALUATION a) "THEOREM": For any sets A, B, and C, if A UB=AU C, then B = C. "Proof": We will prove this by contradiction. Suppose that A UB and A UC are not equal. Then there is some object x that is in one and not the other. We proceed by looking at two cases: First look at the case where x EA UB and not in A U C. Then x is not in A (be- cause, if it were, it would also be in A U C). So x must be in B (otherwise it wouldn't would also be in A U C). be in A U B). Also, x is not in C (because, if it were, Therefore x is in B and not in C, which contradicts the condition B The case where x E AUC and not in A U B is done by a parallel argument. = C.
Transcribed Image Text:(see instructions for Exercise 6 in Section 2.2): PROOF EVALUATION a) "THEOREM": For any sets A, B, and C, if A UB=AU C, then B = C. "Proof": We will prove this by contradiction. Suppose that A UB and A UC are not equal. Then there is some object x that is in one and not the other. We proceed by looking at two cases: First look at the case where x EA UB and not in A U C. Then x is not in A (be- cause, if it were, it would also be in A U C). So x must be in B (otherwise it wouldn't would also be in A U C). be in A U B). Also, x is not in C (because, if it were, Therefore x is in B and not in C, which contradicts the condition B The case where x E AUC and not in A U B is done by a parallel argument. = C.
PROOF EVALUATION (This type of exercise will appear occasionally): Each of the follow- ing is a proposed "proof" of a "theorem". However the "theorem" may not be a true statement, and even if it is, the "proof" may not really be a proof. You should read each "theorem" and "proof" carefully and decide and state whether or not the "theorem" is true. Then: G If the "theorem" is false, find where the "proof" fails. (There has to be some error.) . If the "theorem" is true, decide and state whether or not the "proof" is correct. If it is not correct, find where the "proof" fails.
Transcribed Image Text:PROOF EVALUATION (This type of exercise will appear occasionally): Each of the follow- ing is a proposed "proof" of a "theorem". However the "theorem" may not be a true statement, and even if it is, the "proof" may not really be a proof. You should read each "theorem" and "proof" carefully and decide and state whether or not the "theorem" is true. Then: G If the "theorem" is false, find where the "proof" fails. (There has to be some error.) . If the "theorem" is true, decide and state whether or not the "proof" is correct. If it is not correct, find where the "proof" fails.
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