University of Mumbai 2009-3rd Sem B.Sc Information Technology (IT) Logic, Discrete Mathematical Structures - Question Paper
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Con. 243-09.
( 3 Hours) [ Total Marks : 100
N.B. : (1) Question No. 1 is compulsory.
(2) Attempt any four questions from the remaining questions.
(3) All questions carry equal marks.
1. (a) Find how many integers between 1 and 60 are not divisible by 2, nor by 3, and 10 nor by 5 ?
(b) In a survey of 60 people, it was found that 25 read News week magazine, 26 read 10 Time, and 26 read Fortune. Also 9 read both Newsweek and Fortune. 11 read both Newsweek and Time. 8 read both Time and Fortune and 8 read no magazine at all.
(i) Find the number of people who red all the 3 magazines.
(ii) Determine the number of people who read exactly one magazine.
2. (a) ( P v Q) <-> [ Q v (R -> P) ] 20
(b) [ P A ( P -> Q ) ] -> Q
(c) (P v Q) v - P
(d) ~ ( ~ P v - Q)
3. (a) Define weight of a codeword and find the weights of the following : 10
(i) x = 010010 (ii) x = 111011.
(i) x = 010000 y = 000101
(ii) x = 001100 y = 010110
4. Consider the (3, 8) encoding function e : b3 -> b8 defined by 20
e(000) = 00000000 e(100) = 10100100
e(001) = 10111000 e(101) = 10001001
e(010) = 00101101 e(110) = 00011100
e(011) = 10010101 e(111) = 00110001
How many errors will e detect ?
5. Let A = { 1, 2, 3, 4 } and relation R = { (1, 2), (1, 3), (2, 4), (3, 2) }. Find the transitive 20 closure of R by Warshalls Algorithm.
6. (a) Let A = { 1,2, 3,4,6} and let R be the relation on A defined by x divides y 20
written xy. So that
R = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6) (3, 3), (3, 6), (6, 6) }.
Draw the directed graph of R.
(b) Find the matrix M of the relation R in the problem (6a).
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