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Problem_Set_02

Problem_Set_02 - ArsDigita University Month 2 Discrete...

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ArsDigita University Month 2: Discrete Mathematics - Professor Shai Simonson Problem Set 2 – Sets, Functions, Big-O, Rates of Growth 1. Prove by formal logic: a. The complement of the union of two sets equals the intersection of the complements. b. The complement of the intersection of two sets equals the union of the complements. c. ( B - A ) ( C - A ) = ( B C ) – A . d. If two sets are subsets of each other then they are equal. 2. Generalize De Morgan’s laws for n sets and prove the laws by induction. 3. Prove by induction on the size of the set, that the power set P(A) has cardinality 2 |A | . 4. A B is defined to be the set of all elements in A or B but not in both A and B. a. Determine whether or not is commutative. Prove your answer. b. Determine whether is associative. Prove your answer. c. Determine whether can be distributed over union. Prove your answer. d. Determine whether can be distributed over intersection. Prove your answer. 5. Assume a universal set of 8 elements. Given a set A = a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 , represented by 8 bits, explain how to use bitwise and/or/not operations, in order to: a. Extract the rightmost bit of set A . b. Make the odd numbered bits equal 0. c. Make bits 4-6 equal to 1. Given another set B, explain how to: a. Determine if A B. b. Extract A - B.
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