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Posted Date: 19 Nov 2009 Posted By: Manish Jain Member Level: Gold
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2007 Madras University M.C.A Introduction to Discrete Structures Question paper
Time: Three hours
Maximum: 75 marks
PART A - [5 x 5 = Marks 25]
Answer ALL questions.
All questions carry equal marks.
1. (a) Show the equivalence
(P d Q) t (P Ù Q) v ( P Ù Q)
Or
(b) Show the {Ù,v} {v} and { } are not functionally complete.
2. (a) Let Z be the set of integers and let R be the relation called "congruence modulo 3" defined by R = {x,y Î|x Î Z Ù y Î Z (x - y) is divisible by 3}. Determine the equivalence classes generated by the elements of Z.
Or
(b) Give an example of a set X such that r( ), x Í is a totally ordered set.
3. (a) State and prove Langrange's theorem.
Or
(b) Prove that every subgroup of a cyclic group is cyclic.
4. (a) Explain the method of finding the adjacency matrix.
Or
(b) How will you represent a graph using arrays.
5. (a) Explain the term 'right derivation' using a grammar to generate a language.
Or
(b) Give a context-free grammar which generates L = {w / w contains twice as many 0s as 1s}
PART B - [5 x 10 = Marks 50]
Answer any FIVE questions.
All questions carry equal marks.
6. Show that (x) (P (x) v Q (x)) r(x) P (x) v ( Fx) Q (x).
7. show that R Ù (P v Q) is a valid conclusion from the premises P v Q, Q 2 R, P 2 M and M.
8. Show that addition of matrics form an abelian group.
9. Prove that for any commutative monoid M,* , the set of idempotent element s of M forms a submonoid.
10. Show that for any two sets A and B A -(A Ç B) = A - B.
11. (a) Show that the set of divisors of a positive integer n is recursive.
(b) Show that the sets of even and odd natural numbers are both recursive.
12. Contrast a context-sensitive grammar for the language { } w w/ *{} {a,b,c '-Ù, where w contains the same number of a's, b's and c's.
13. Explain the terms 'sentential form', regular grammar and grammars.
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