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Posted By: SRIMATHI Member Level: Diamond Posted Date: 08 May 2008
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2005 Indira Gandhi National Open University (IGNOU) Post Graduate Diploma Computer Application CS07 : Discrete Mathematics June, 2005 Question paper
PGDCA \ MCA (II Yr)
Term-End Examination
June, 2005
CS07: Discrete Mathematics
Time: 3 hours
Maximum Marks: 75
Note : Question No. 1 is compulsory. Answer any three questions from the rest.
1. (a) (a) Which of the following are statements ? Give reasons for your answer. (4)
(i) What a lovely day ! (ii) 2 + 3 = 6 (iii) 2 + x = 7 (iv) What is the time now ?
(b) Give an example of a finite linearly ordered set. You have to verify all the properties of a linearly ordered set for your example. (4)
(c) Find the product G1 x G2 of the following graphs :
-----DIAGRAM-----
Give a cycle of length 6 in G1 x G2, if one exists (4)
(d) Let U = {1, 2, 3, 4, 5, 6,7}, A = {1,2 ,7}, B = {1, 2, 3, 4, 5}, C = {4, 6, 7}.
(i) Findnd A-B and B-C.
(ii) Check that (A U B) n C = (A n C) U (B n C).
(iii) State and verify any one of the De Morgan's laws for A and B. (4)
(e) Write the equivalent form of the following using Predicate Calculus : (4)
(i) Every MCA graduate knows either C++ programming of Java programming.
(ii) There are web programmers who know PERL but not PHP.
(f) Let A = {1, 2, 3} and R1 and R2 be relations on A given by
R1 = {(1, 1), (1, 2), (2,3)}
R2 = {{1, 3), (2, 1), (1, 1), (3, 1)}
(i) Write the relation matrices of R1 and R2.
(ii) Find the relation matrix of R1R2.
(iii) Check wheiher R1R2 is an equivalence relation. (3)
(g) For the lattice (D(231), g.c.d., l.c.m.) draw the Hasse diagram. Also check that the lattice is complemented. (3)
(h) Find the diameter and radius of the following graph : (4)
-----DIAGRAM-----
2. (a) Determine if the following system specifications are consistent using a truth table. (6)
'The diagnostic message is stored in the buffer or it is retransmitted'
'The diagnostic message is not stored in the buffer'
'If the diagnostic message is stored in the buffer it is retransmitted'
(b) Let f = 1 2 3 4 5 and
2 3 1 5 4
g = 1 2 3 4 5
3 4 2 5 1What is f-1
Write f o g as a product of disJoint cycles. Is f o g even? Why? (5)
(c) Find the binary tree representation of (x + y) (7x - xy). (4)
3. (a) The adjacency rnatrix of a graph G is
0 0 1 1 0
0 0 1 1 1
A = 1 1 0 0 1
1 1 0 0 0
0 1 1 0 0
Find A + A2 + A3 +A4. Is G connected? Check your answer by drawing the graph G. (6)
(b) Check that (P v Q) ? (?P ? Q) is a tautology. (5)
(c) Draw the switching circuit corresponding to the function ((x + y) . (yz)) + (x+z). (4)
4. (a) ln a survey, 80 Computer Science teachers were asked which of the three programming languages, Pascal, Fortran and C, have they used to introduce programming to students- 40 teachers said they have used Pascal, 35 said they have used Fortran, 25 said they have used C, 20 said they have used both Pascal and Fortran, 17 said they have used Pascal and C, 5 said they have used Fortran and C, and 3 said they have used all the three languages.
(i) How many teachers have used Fortran alone ?
(ii) How many teachers have used both Pascal and C ?
(iii) How rnany teachers have used Fortran, but not C ?
(iv) How many used none of the three languages? (6)
(b) Apply Dijktra's algorithm to find the shortest path between a and Z in the following weighted graph. (7)
-----DIAGRAM-----
(c) Express (P ^ Q) v (Q ^ R) in disjunctive normal form. (2)
5. (a) Simplify the following boolean expression using a Karnaugh map : (7) x1x2x3xBAR;4 + x1xBAR2x3x4 + xBAR1xBAR2x3x4 + xBAR1x2xBAR3x4 + x1x2xBAR3xBAR4
(b) Write -92 and 120 in 1's complement and use this to find 120 - 92 in 1's complement form. (3)
(c) Let X = {1, 2, 3, 4, 5, 6}.
A = {0/1, 0.4/2, 0.3/3, 0.5/4, 0.1/5, 1/6}
B = {0.1/1, 0.3/2, 0.1/3, 0.1/4, 0.2/5, 0.1/6}
(i) Which of the fuzzy sets A and B are normal ? Give reasons for your answer.
(ii) Find A n B. (iii) Find Ac. (iv) Find the support ol A. (v) Find A0.1. (5)
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