(15pts) Let A, B and C be sets. Determine whether or not the following are valid. Justify your answer by using either set identities or membership tables. You can also use a counterexample to show that two sets are not equivalent. Notice, that the difference between two sets A and B can be denoted A\B or A-B. A sample solution is provided in part (f). (a) An (B-C) = (A-C) NB (b) A - (B-C) = (A - B) - C (c) (A-B) U (B-A) = (An B) (d) (A-B) U (B-A) = AUB (e) ((A-B) UCU(AUB)) UB=U, where U is the Universal set. (f) ((ANB)u(ANB)) n((BnB) − A) = 0 Solution: The equation is true. We will show this using set identities. ((ANB) u (AnB)) n ((BnB) -A) (An (BUB)) n((BnB) nĀ) (An (BUB)) n((BUB) NA) (ANU)n (UNA) ANĀ 0 Distributive Law, Difference Equivalence De Morgan's Law, Complementation Law Complement Law (Law of Excluded Middle) (twice) Identity (twice) Complement Law (Contradiction)
(15pts) Let A, B and C be sets. Determine whether or not the following are valid. Justify your answer by using either set identities or membership tables. You can also use a counterexample to show that two sets are not equivalent. Notice, that the difference between two sets A and B can be denoted A\B or A-B. A sample solution is provided in part (f). (a) An (B-C) = (A-C) NB (b) A - (B-C) = (A - B) - C (c) (A-B) U (B-A) = (An B) (d) (A-B) U (B-A) = AUB (e) ((A-B) UCU(AUB)) UB=U, where U is the Universal set. (f) ((ANB)u(ANB)) n((BnB) − A) = 0 Solution: The equation is true. We will show this using set identities. ((ANB) u (AnB)) n ((BnB) -A) (An (BUB)) n((BnB) nĀ) (An (BUB)) n((BUB) NA) (ANU)n (UNA) ANĀ 0 Distributive Law, Difference Equivalence De Morgan's Law, Complementation Law Complement Law (Law of Excluded Middle) (twice) Identity (twice) Complement Law (Contradiction)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:(15pts) Let A, B and C be sets. Determine whether or not the following are valid. Justify your answer by
using either set identities or membership tables. You can also use a counterexample to show that two sets are
not equivalent. Notice, that the difference between two sets A and B can be denoted A\B or A-B. A sample
solution is provided in part (f).
(a) An (B-C) = (A-C) NB
(b) A - (B-C) = (A - B) - C
(c) (A-B) U (B-A) = (An B)
(d) (A-B) U (B-A) = AUB
(e) ((A-B) UCU(AUB)) UB=U, where U is the Universal set.
(f) ((ANB)u(ANB)) n((BnB) − A) = 0
Solution: The equation is true. We will show this using set identities.
((ANB) u (AnB)) n ((BnB) -A)
(An (BUB)) n((BnB) nĀ)
(An (BUB)) n((BUB) NA)
(ANU)n (UNA)
ANĀ
0
Distributive Law, Difference Equivalence
De Morgan's Law, Complementation Law
Complement Law (Law of Excluded Middle) (twice)
Identity (twice)
Complement Law (Contradiction)
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