Acharya Nagarjuna University (ANU) 2006 M.C.A DISCRETE MATHEMATICS - Question Paper
M.C.A. (Previous) Degree Examination, May 2006
Paper - VIII - DiscRETE MATHEMATICS
(DMCA 108)
8
M.C.A.(Previous) DEGREE EXAMINATION, MAY 2006 PAPER -VIII - DISCRETE MATHEMATICS
Time: Three hours Maximum: 75 marks
SECTION - A - (3 X 15 = 45 marks)
Answer any THREE of the following
1. (a) Test the validity of the following argument.
If I study, then I will not fail mathematics If I do not play basket ball, then I will study.
But I failed Mathematics
Therefore I must have played basket ball.
(b) Find the truth tables for (i) p v -,g (ii) -,pA-,q
2. (a) For any ne.V, Let D =(0,1/ft), the open interval from O to 1. Find :
(i)Z)3DT (ii) Ds n.D,c (iii) Z)suDf (iv)DsnDf.
(b) Let f: R * R be defined by /(x) = 2*-3 . Now f is One-to-one and Onto: hence / has an inverse function f~l. Find a formula for /"].
3. (a) Show that every finite semigroup has an idempotent.
(b)Let (5,*) be a semigroup and Ze5 be a left zero. Show that for any jr e $,x*Z is also a left zero.
4. (a) Ex plain the p ro p ert ie s of b in ary op erat io n s.
(b) Determine whether J?j = {0,5,10} and H-, = {0.4,8,12} are subgroups of Z1S.
5. Let _Y= {ls2,3s.....} be ordered by divisibility. State whether each of the following subsets of Ware
linearly ordered. (a) {24,2,6} (b) _ = {l,2,3} (c) {7} (d) {3,15,5} (e) {2,8,32,4}
{f) {l5,530}
SECTION - B - (5 X 5 = 25 marks)
Answer any FIVE of the following
6. Negate each of the following statements:
(a) If the teacher is absent, then some students do not complete their homework.
(b) All the students completed their homework and the teacher is present.
(c) Some of the students did not complete their homework or the teacher is absent.
7. Construct the truth table for [Qj q) ap] > q .
8. Consider the group G ={1,2,3,4,5:61 under multiplication modulo 7.
(a) Find the multiplication table of G.
(b) Find 2-1,3-',(TL.
(c) Find the orders and subgroups generated by 2 and 3.
(d) Is G cyclic.
9. Define well-ordered sets with example.
10. Suppose P(p) = + o1h + 2 + ... + amnm has degree in. Prove P(n) = 0(nm) .
11. Explain conjunction and disjunction normal forms.
12. Solve the recurrence relation an a = + 3ji2 + 3w +1 where ac =1 by method of substitution.
13. Using arithmetic modulo m=15, evaluate:
(a) 9 4 13
(b) 7 + 11
(c) 4-9
(d) 2 - 10
SECTION - C - (5 X 1 = 5 marks]
Answer any ALL of the following
14. What is symmetric relation?
15. What is composition function?
16. W h at is H o m oiriorph ism s?
17. Define submonoid.
18. What is subalgebra?
A A * * *
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