questionbankunit1.pdf - Question Bank Unit 1 1 Let U={1,2,3,4,5,6,7,8,9 be the universal set A B and C are three sets A ={1,3,5,7,9 B={2,3,4,5 C={5,6,7

questionbankunit1.pdf - Question Bank Unit 1 1 Let...

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Question Bank: Unit 1 1. Let U={1,2,3,4,5,6,7,8,9} be the universal set. A, B, and C are three sets: A = {1,3,5,7,9}, B={2,3,4,5}, C={5,6,7}. Please explicitly show the members of the following sets. (i) A B, (ii) A B, (iii) The complement of A, (iv) A-C, (v) C-A, (vi) C (A B), (vii) (C A) (C B) (viii) C (A B), (ix) (C A) (C B) 2. (i) Given any two sets A and B, is it true that the set A-B is always equal to the intersection of A and the complement of B. If true, explain why it is true. If not, give a counter example. (ii) Please apply the distributive laws of set operations and the observation in (i) to prove that the set (X Y)- Z is always equal to the set (X-Z) (Y-Z) given any sets X, Y, and Z. 3. Given any 4 sets A, B, C, and D, is it true that (A C) (A D) (B C) (B D) is always equal to (A B) (C D)? If true, use the distributive laws, the associative laws, and the idempotent laws to prove it. If not, give a counter example. Use mathematical induction to prove the following statements: 4. For ev ery integer n≥1, ( ∑ 1≤ i ≤ n i ) = n(n+1)/2. In other words, the summation of the first n positive integers equals n(n+1)/2. 5. For every integer n≥1, ( ∑ 1≤ i ≤ n i 3 ) = (n(n+1)/2) 2 . In other words, the summation of the cubes of the first n positive integers equals the square of the summation of the first n positive integers. 6. For every integer n≥0, ( ∑ 0 ≤ i ≤ n r i ) = ( r n+1 1) / (r 1) where r is some real number and r ≠ 1. 7. For every integer n ≥ 5, we have 2 n > n 2 8. Consider three specific sets X={1,2}, Y={a, b, c}, Z={b, c, d}. Show what are the following sets. (Please explicitly list the members of these sets). (i) Power (X), i.e. the power set of X, (ii) X Y, (iii) X (Y Z) and (iv) (X Y) ( X Z) 9. Consider any three sets X, Y, and Z. (i) Is X (Y Z) always equal to (X Y) ( X Z)? (ii) Is X (Y Z) always equal to (X Y) ( X Z)? (iii) Is Power (X Y) always equal to Power (X) Power (Y) (iv) Is Power (X Y) always equal to Power (X) Power (Y) 10 . Consider the set A={p, q, r} and the two binary relations R={ (p, q), (q, r), (r, p) } and S= {(q,p) (r,q), (p, r) } on A. do the followings: (i) Put down the matrix representations of R and S respectively.
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