5. (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) B (b) A (B-C) = (A - B) - C (c) (A - B) U (B − A) = (ANB) (d) (A - B) U (B − A) = AUB (e) ((A-B)UCU (AUB)) UB= U, where U is the Universal set. ((ANB)u(ANB))n((B^B) − A) = Ø) (f) Solution: The equation is true. We will show this using set identities. ((An B) u (An B)) n ((BnB) - A) (An (BUB)) n ((BnB) nA) (An (BUB)) n ((BUB) nA) (ANU)n(UNĀ) AnÃ Ø 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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